C3 - Homogeneous second-order linear ODE


Example 1 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 2 \, {y''} = -24 \, {y'} - 72 \, {y} \]

  2. \[ -4 \, {x'} - 34 \, {x} = 2 \, {x''} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)} \]

\[ {y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)} \]


Example 2 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -87 \, {x} = -30 \, {x'} + 3 \, {x''} \]

  2. \[ -40 \, {y'} = 2 \, {y''} + 200 \, {y} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(2 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 5\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(5 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)} \]


Example 3 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 3 \, {x''} = 30 \, {x'} - 75 \, {x} \]

  2. \[ 6 \, {y} - 6 \, {y'} = -3 \, {y''} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t} \]

\[ {x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)} \]


Example 4 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 0 = 200 \, {x} + 2 \, {x''} + 40 \, {x'} \]

  2. \[ 0 = -12 \, {y'} - 26 \, {y} - 2 \, {y''} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(2 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 3\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-3 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)} \]


Example 5 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 0 = -3 \, {y''} + 6 \, {y'} - 78 \, {y} \]

  2. \[ 3 \, {x''} = -108 \, {x} + 36 \, {x'} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(5 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 1\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{t} \]

\[ {x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)} \]


Example 6 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -8 \, {x'} = 8 \, {x} + 2 \, {x''} \]

  2. \[ 12 \, {y'} + 3 \, {y''} = -24 \, {y} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} \]


Example 7 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 6 \, {x'} = -30 \, {x} - 3 \, {x''} \]

  2. \[ 3 \, {y} = -3 \, {y''} - 6 \, {y'} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(3 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 1\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-t\right)} \]

\[ {y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)} \]


Example 8 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -3 \, {y''} - 123 \, {y} = 24 \, {y'} \]

  2. \[ -2 \, {x''} = 28 \, {x'} + 98 \, {x} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(5 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 4\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-4 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)} \]


Example 9 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -2 \, {x''} - 20 \, {x'} = 50 \, {x} \]

  2. \[ -58 \, {y} = 20 \, {y'} + 2 \, {y''} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)} \]


Example 10 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 68 \, {x} + 12 \, {x'} + 2 \, {x''} = 0 \]

  2. \[ 12 \, {y} + 12 \, {y'} = -3 \, {y''} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} \]


Example 11 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -28 \, {y'} = 2 \, {y''} + 98 \, {y} \]

  2. \[ 20 \, {x'} = 58 \, {x} + 2 \, {x''} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(2 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 5\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(5 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)} \]


Example 12 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -3 \, {x''} - 60 \, {x} = -24 \, {x'} \]

  2. \[ -4 \, {y'} + 2 \, {y''} = -2 \, {y} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)} \]

\[ {y} = k_{1} t e^{t} + k_{2} e^{t} \]


Example 13 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 24 \, {y'} = 60 \, {y} + 3 \, {y''} \]

  2. \[ 16 \, {x'} - 32 \, {x} = 2 \, {x''} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)} \]


Example 14 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 0 = -2 \, {y''} + 20 \, {y'} - 58 \, {y} \]

  2. \[ 6 \, {x'} = 3 \, {x''} + 3 \, {x} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(2 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 5\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(5 \, t\right)} \]

\[ {x} = k_{1} t e^{t} + k_{2} e^{t} \]


Example 15 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 0 = 4 \, {x'} + 2 \, {x} + 2 \, {x''} \]

  2. \[ 6 \, {y'} = 3 \, {y''} + 78 \, {y} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(5 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 1\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{t} \]

\[ {x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)} \]


Example 16 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -34 \, {y} + 16 \, {y'} = 2 \, {y''} \]

  2. \[ -2 \, {x''} - 8 \, {x} = 8 \, {x'} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(i - 4\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(4 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} \]


Example 17 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 0 = 3 \, {y''} - 30 \, {y'} + 150 \, {y} \]

  2. \[ 108 \, {x} + 36 \, {x'} = -3 \, {x''} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(5 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 5\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(5 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)} \]


Example 18 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -2 \, {y''} - 36 \, {y'} = 162 \, {y} \]

  2. \[ 0 = -39 \, {x} + 12 \, {x'} - 3 \, {x''} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(3 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 2\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(2 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)} \]


Example 19 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -128 \, {x} - 32 \, {x'} - 2 \, {x''} = 0 \]

  2. \[ -18 \, {y'} + 3 \, {y''} = -39 \, {y} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)} \]


Example 20 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -2 \, {y''} - 2 \, {y} = -4 \, {y'} \]

  2. \[ 3 \, {x''} = -24 \, {x'} - 51 \, {x} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(i + 4\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-4 \, t\right)} \]

\[ {y} = k_{1} t e^{t} + k_{2} e^{t} \]


Example 21 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 36 \, {x} + 2 \, {x''} + 12 \, {x'} = 0 \]

  2. \[ -192 \, {y} + 48 \, {y'} = 3 \, {y''} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(3 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 3\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-3 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)} \]


Example 22 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -2 \, {x''} + 20 \, {x'} = 50 \, {x} \]

  2. \[ 18 \, {y'} = -3 \, {y''} - 54 \, {y} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(3 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 3\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-3 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)} \]


Example 23 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -24 \, {x'} = 123 \, {x} + 3 \, {x''} \]

  2. \[ -32 \, {y'} + 2 \, {y''} = -128 \, {y} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(5 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 4\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-4 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)} \]


Example 24 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 0 = -12 \, {y'} - 18 \, {y} - 2 \, {y''} \]

  2. \[ 12 \, {x'} - 2 \, {x''} = 68 \, {x} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(5 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 3\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(3 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} \]


Example 25 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 50 \, {y} + 20 \, {y'} = -2 \, {y''} \]

  2. \[ -18 \, {x'} + 39 \, {x} = -3 \, {x''} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)} \]


Example 26 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 32 \, {x} + 2 \, {x''} = -16 \, {x'} \]

  2. \[ -24 \, {y} = 12 \, {y'} + 3 \, {y''} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} \]


Example 27 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 2 \, {y} + 4 \, {y'} = -2 \, {y''} \]

  2. \[ 2 \, {x''} + 82 \, {x} = 20 \, {x'} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)} \]


Example 28 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -50 \, {y} = -16 \, {y'} + 2 \, {y''} \]

  2. \[ 3 \, {x''} + 192 \, {x} = -48 \, {x'} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)} \]


Example 29 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 20 \, {y'} + 2 \, {y''} = -58 \, {y} \]

  2. \[ 8 \, {x'} + 8 \, {x} + 2 \, {x''} = 0 \]

Answer:

\[ {y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} \]


Example 30 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -2 \, {x''} = 200 \, {x} + 40 \, {x'} \]

  2. \[ 3 \, {y''} = -24 \, {y'} - 75 \, {y} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(3 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 4\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-4 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)} \]


Example 31 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 3 \, {x''} = -42 \, {x'} - 147 \, {x} \]

  2. \[ -40 \, {y} - 2 \, {y''} - 16 \, {y'} = 0 \]

Answer:

\[ {y} = c_{1} e^{\left(\left(2 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 4\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-4 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)} \]


Example 32 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 6 \, {y'} - 3 \, {y} - 3 \, {y''} = 0 \]

  2. \[ 12 \, {x'} = -2 \, {x''} - 68 \, {x} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)} \]

\[ {y} = k_{1} t e^{t} + k_{2} e^{t} \]


Example 33 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -48 \, {x} = 24 \, {x'} + 3 \, {x''} \]

  2. \[ 3 \, {y''} + 24 \, {y'} = -75 \, {y} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(3 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 4\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-4 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} \]


Example 34 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -12 \, {y'} - 2 \, {y''} - 18 \, {y} = 0 \]

  2. \[ 0 = 16 \, {x'} - 2 \, {x''} - 82 \, {x} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(5 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 4\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(4 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} \]


Example 35 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -24 \, {y'} - 3 \, {y''} = 60 \, {y} \]

  2. \[ 2 \, {x} + 4 \, {x'} = -2 \, {x''} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(2 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 4\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-4 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)} \]


Example 36 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 24 \, {x} - 12 \, {x'} = -3 \, {x''} \]

  2. \[ 12 \, {y'} = 18 \, {y} + 2 \, {y''} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(2 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 2\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(2 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)} \]


Example 37 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -3 \, {x''} - 108 \, {x} = -36 \, {x'} \]

  2. \[ 100 \, {y} - 20 \, {y'} = -2 \, {y''} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(5 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 5\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(5 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)} \]


Example 38 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -20 \, {y} + 4 \, {y'} = 2 \, {y''} \]

  2. \[ 0 = 2 \, {x''} + 98 \, {x} + 28 \, {x'} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t} \]

\[ {x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)} \]


Example 39 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 3 \, {y''} + 27 \, {y} - 18 \, {y'} = 0 \]

  2. \[ -78 \, {x} - 3 \, {x''} = 30 \, {x'} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(i + 5\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-5 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)} \]


Example 40 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 8 \, {x} + 8 \, {x'} = -2 \, {x''} \]

  2. \[ -2 \, {y''} + 16 \, {y'} = 40 \, {y} \]

Answer:

\[ {y} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)} \]

\[ {y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)} \]

\[ {x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} \]


Example 41 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -34 \, {x} - 2 \, {x''} = 16 \, {x'} \]

  2. \[ -4 \, {y'} = -2 \, {y} - 2 \, {y''} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(i + 4\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-4 \, t\right)} \]

\[ {y} = k_{1} t e^{t} + k_{2} e^{t} \]


Example 42 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 78 \, {x} - 30 \, {x'} = -3 \, {x''} \]

  2. \[ 0 = -2 \, {y''} - 128 \, {y} - 32 \, {y'} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)} \]


Example 43 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -3 \, {y''} - 12 \, {y'} = 12 \, {y} \]

  2. \[ 52 \, {x} + 2 \, {x''} = 20 \, {x'} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} \]


Example 44 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 2 \, {y} + 2 \, {y''} = -4 \, {y'} \]

  2. \[ 3 \, {x''} = 6 \, {x'} - 78 \, {x} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(5 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 1\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{t} \]

\[ {y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)} \]


Example 45 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -24 \, {x} = 3 \, {x''} - 12 \, {x'} \]

  2. \[ -72 \, {y} + 24 \, {y'} = 2 \, {y''} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(2 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 2\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(2 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)} \]


Example 46 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -54 \, {y'} = -3 \, {y''} - 243 \, {y} \]

  2. \[ -50 \, {x} = -12 \, {x'} + 2 \, {x''} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(4 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 3\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(3 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(9 \, t\right)} + k_{2} e^{\left(9 \, t\right)} \]


Example 47 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 12 \, {x'} = 3 \, {x''} + 39 \, {x} \]

  2. \[ 0 = 3 \, {y''} - 36 \, {y'} + 108 \, {y} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(3 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 2\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(2 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)} \]


Example 48 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 2 \, {y''} + 4 \, {y'} + 2 \, {y} = 0 \]

  2. \[ -24 \, {x'} = 3 \, {x''} + 96 \, {x} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(4 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 4\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-4 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)} \]


Example 49 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ -28 \, {y'} = 98 \, {y} + 2 \, {y''} \]

  2. \[ 0 = 39 \, {x} - 18 \, {x'} + 3 \, {x''} \]

Answer:

\[ {x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)} \]


Example 50 πŸ”—

Explain how to find the general solution to each given ODE using exponential functions.

For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.

  1. \[ 2 \, {x''} + 20 \, {x} = -12 \, {x'} \]

  2. \[ -2 \, {y''} - 24 \, {y'} - 72 \, {y} = 0 \]

Answer:

\[ {x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)} \]

\[ {x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)} \]

\[ {y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)} \]