## C3 - Homogeneous second-order linear ODE

#### Example 1 π

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

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

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

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

#### Example 2 π

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

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

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

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

#### Example 3 π

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. $6 \, {y} - 6 \, {y'} = -3 \, {y''}$

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

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

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

#### Example 4 π

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

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

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

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

#### Example 5 π

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. $3 \, {x''} = -108 \, {x} + 36 \, {x'}$

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

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

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

#### Example 6 π

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. $12 \, {y'} + 3 \, {y''} = -24 \, {y}$

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

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

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

#### Example 7 π

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. $3 \, {y} = -3 \, {y''} - 6 \, {y'}$

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

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

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

#### Example 8 π

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

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

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

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

#### Example 9 π

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

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

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

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

#### Example 10 π

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. $12 \, {y} + 12 \, {y'} = -3 \, {y''}$

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

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

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

#### Example 11 π

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

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

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

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

#### Example 12 π

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

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

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

${y} = k_{1} t e^{t} + k_{2} e^{t}$

#### Example 13 π

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

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

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

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

#### Example 14 π

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. $6 \, {x'} = 3 \, {x''} + 3 \, {x}$

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

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

${x} = k_{1} t e^{t} + k_{2} e^{t}$

#### Example 15 π

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. $6 \, {y'} = 3 \, {y''} + 78 \, {y}$

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

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

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

#### Example 16 π

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. $-2 \, {x''} - 8 \, {x} = 8 \, {x'}$

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

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

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

#### Example 17 π

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. $108 \, {x} + 36 \, {x'} = -3 \, {x''}$

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

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

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

#### Example 18 π

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. $0 = -39 \, {x} + 12 \, {x'} - 3 \, {x''}$

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

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

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

#### Example 19 π

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. $-18 \, {y'} + 3 \, {y''} = -39 \, {y}$

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

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

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

#### Example 20 π

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. $3 \, {x''} = -24 \, {x'} - 51 \, {x}$

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

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

${y} = k_{1} t e^{t} + k_{2} e^{t}$

#### Example 21 π

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. $-192 \, {y} + 48 \, {y'} = 3 \, {y''}$

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

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

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

#### Example 22 π

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. $18 \, {y'} = -3 \, {y''} - 54 \, {y}$

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

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

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

#### Example 23 π

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

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

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

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

#### Example 24 π

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

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

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

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

#### Example 25 π

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. $-18 \, {x'} + 39 \, {x} = -3 \, {x''}$

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

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

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

#### Example 26 π

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. $-24 \, {y} = 12 \, {y'} + 3 \, {y''}$

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

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

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

#### Example 27 π

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

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

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

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

#### Example 28 π

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. $3 \, {x''} + 192 \, {x} = -48 \, {x'}$

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

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

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

#### Example 29 π

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

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

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

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

#### Example 30 π

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. $3 \, {y''} = -24 \, {y'} - 75 \, {y}$

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

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

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

#### Example 31 π

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

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

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

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

#### Example 32 π

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

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

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

${y} = k_{1} t e^{t} + k_{2} e^{t}$

#### Example 33 π

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. $3 \, {y''} + 24 \, {y'} = -75 \, {y}$

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

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

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

#### Example 34 π

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

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

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

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

#### Example 35 π

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

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

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

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

#### Example 36 π

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

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

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

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

#### Example 37 π

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

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

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

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

#### Example 38 π

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

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

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

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

#### Example 39 π

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. $-78 \, {x} - 3 \, {x''} = 30 \, {x'}$

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

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

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

#### Example 40 π

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

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

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

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

#### Example 41 π

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. $-4 \, {y'} = -2 \, {y} - 2 \, {y''}$

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

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

${y} = k_{1} t e^{t} + k_{2} e^{t}$

#### Example 42 π

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

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

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

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

#### Example 43 π

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

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

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

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

#### Example 44 π

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. $3 \, {x''} = 6 \, {x'} - 78 \, {x}$

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

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

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

#### Example 45 π

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

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

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

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

#### Example 46 π

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

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

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

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

#### Example 47 π

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. $0 = 3 \, {y''} - 36 \, {y'} + 108 \, {y}$

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

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

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

#### Example 48 π

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. $-24 \, {x'} = 3 \, {x''} + 96 \, {x}$

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

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

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

#### Example 49 π

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. $0 = 39 \, {x} - 18 \, {x'} + 3 \, {x''}$

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

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

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

#### Example 50 π

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

${x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}$
${x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}$
${y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}$