C2 - Non-homogeneous first-order linear ODE


Example 1 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -40 \, e^{\left(4 \, t\right)} \sin\left(5 \, t\right) + 4 \, {y'} = 16 \, {y} \]

Answer:

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


Example 2 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -24 \, e^{\left(2 \, t\right)} = 20 \, {y} - 4 \, {y'} \]

Answer:

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


Example 3 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 12 \, \cos\left(2 \, t\right) e^{\left(-t\right)} - 3 \, {y'} = 3 \, {y} \]

Answer:

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


Example 4 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -4 \, {y'} + 8 \, e^{\left(-5 \, t\right)} = 16 \, {y} \]

Answer:

\[ {y} = k e^{\left(-4 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \]


Example 5 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 0 = -{y'} - {y} + 10 \, e^{\left(4 \, t\right)} \]

Answer:

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


Example 6 πŸ”—

Explain how to find the general solution to the given ODE.

\[ {y} = 3 \, \cos\left(-t\right) e^{t} + {y'} \]

Answer:

\[ {y} = k e^{t} + 3 \, e^{t} \sin\left(-t\right) \]


Example 7 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 4 \, {y'} - 16 \, {y} + 64 \, e^{\left(-4 \, t\right)} = 0 \]

Answer:

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


Example 8 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 0 = 2 \, {y'} + 10 \, {y} - 4 \, e^{\left(-5 \, t\right)} \]

Answer:

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


Example 9 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 0 = -24 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right) + 15 \, {y} + 3 \, {y'} \]

Answer:

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


Example 10 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 10 \, {y} - 2 \, {y'} = 6 \, e^{\left(5 \, t\right)} \]

Answer:

\[ {y} = k e^{\left(5 \, t\right)} - 3 \, t e^{\left(5 \, t\right)} \]


Example 11 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 3 \, {y'} + 6 \, e^{t} = 3 \, {y} \]

Answer:

\[ {y} = k e^{t} - 2 \, t e^{t} \]


Example 12 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -{y'} - {y} = -15 \, e^{\left(-t\right)} \sin\left(5 \, t\right) \]

Answer:

\[ {y} = k e^{\left(-t\right)} - 3 \, \cos\left(5 \, t\right) e^{\left(-t\right)} \]


Example 13 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 2 \, \cos\left(-t\right) e^{\left(3 \, t\right)} + {y'} = 3 \, {y} \]

Answer:

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


Example 14 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 6 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right) + {y'} = 5 \, {y} \]

Answer:

\[ {y} = k e^{\left(5 \, t\right)} + 3 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)} \]


Example 15 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 0 = -6 \, {y} - 2 \, {y'} - 30 \, e^{\left(2 \, t\right)} \]

Answer:

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


Example 16 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 0 = 15 \, \cos\left(5 \, t\right) e^{\left(-2 \, t\right)} - 2 \, {y} - {y'} \]

Answer:

\[ {y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(5 \, t\right) \]


Example 17 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 12 \, e^{\left(-4 \, t\right)} \sin\left(4 \, t\right) + 4 \, {y} + {y'} = 0 \]

Answer:

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


Example 18 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -20 \, \cos\left(5 \, t\right) e^{\left(-5 \, t\right)} + 10 \, {y} = -2 \, {y'} \]

Answer:

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


Example 19 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -10 \, \cos\left(5 \, t\right) e^{\left(-4 \, t\right)} + {y'} + 4 \, {y} = 0 \]

Answer:

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


Example 20 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 2 \, {y'} = -6 \, {y} - 42 \, e^{\left(4 \, t\right)} \]

Answer:

\[ {y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(4 \, t\right)} \]


Example 21 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -3 \, {y'} = 15 \, {y} + 9 \, e^{\left(-5 \, t\right)} \]

Answer:

\[ {y} = k e^{\left(-5 \, t\right)} - 3 \, t e^{\left(-5 \, t\right)} \]


Example 22 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -8 \, {y} = -24 \, e^{\left(2 \, t\right)} \sin\left(3 \, t\right) - 4 \, {y'} \]

Answer:

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


Example 23 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -6 \, e^{\left(-4 \, t\right)} \sin\left(2 \, t\right) + 4 \, {y} = -{y'} \]

Answer:

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


Example 24 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -12 \, \cos\left(-4 \, t\right) e^{t} + {y} - {y'} = 0 \]

Answer:

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


Example 25 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 0 = -16 \, {y} + 4 \, {y'} + 12 \, e^{\left(4 \, t\right)} \]

Answer:

\[ {y} = k e^{\left(4 \, t\right)} - 3 \, t e^{\left(4 \, t\right)} \]


Example 26 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -36 \, \cos\left(4 \, t\right) e^{\left(5 \, t\right)} + 3 \, {y'} - 15 \, {y} = 0 \]

Answer:

\[ {y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(4 \, t\right) \]


Example 27 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 24 \, \cos\left(-4 \, t\right) e^{\left(2 \, t\right)} - 3 \, {y'} + 6 \, {y} = 0 \]

Answer:

\[ {y} = k e^{\left(2 \, t\right)} - 2 \, e^{\left(2 \, t\right)} \sin\left(-4 \, t\right) \]


Example 28 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -54 \, e^{\left(4 \, t\right)} = 6 \, {y} + 3 \, {y'} \]

Answer:

\[ {y} = k e^{\left(-2 \, t\right)} - 3 \, e^{\left(4 \, t\right)} \]


Example 29 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 12 \, {y} = 40 \, e^{\left(3 \, t\right)} \sin\left(-5 \, t\right) + 4 \, {y'} \]

Answer:

\[ {y} = k e^{\left(3 \, t\right)} - 2 \, \cos\left(-5 \, t\right) e^{\left(3 \, t\right)} \]


Example 30 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -5 \, {y} = -4 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right) - {y'} \]

Answer:

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


Example 31 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 2 \, {y'} = 4 \, e^{\left(-2 \, t\right)} \sin\left(-t\right) - 4 \, {y} \]

Answer:

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


Example 32 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -24 \, \cos\left(2 \, t\right) e^{\left(-4 \, t\right)} + 16 \, {y} = -4 \, {y'} \]

Answer:

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


Example 33 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 4 \, {y'} - 12 \, {y} - 16 \, e^{\left(5 \, t\right)} = 0 \]

Answer:

\[ {y} = k e^{\left(3 \, t\right)} + 2 \, e^{\left(5 \, t\right)} \]


Example 34 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -4 \, e^{t} = -4 \, {y} + 2 \, {y'} \]

Answer:

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


Example 35 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -8 \, {y} + 4 \, {y'} = 12 \, e^{\left(3 \, t\right)} \]

Answer:

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


Example 36 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -{y} = -2 \, e^{\left(-t\right)} \sin\left(t\right) + {y'} \]

Answer:

\[ {y} = k e^{\left(-t\right)} - 2 \, \cos\left(t\right) e^{\left(-t\right)} \]


Example 37 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -12 \, \cos\left(2 \, t\right) e^{\left(4 \, t\right)} - 3 \, {y'} = -12 \, {y} \]

Answer:

\[ {y} = k e^{\left(4 \, t\right)} - 2 \, e^{\left(4 \, t\right)} \sin\left(2 \, t\right) \]


Example 38 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -4 \, {y'} = 4 \, {y} - 8 \, e^{\left(-t\right)} \]

Answer:

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


Example 39 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 6 \, {y} + 2 \, {y'} = 12 \, \cos\left(2 \, t\right) e^{\left(-3 \, t\right)} \]

Answer:

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


Example 40 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -16 \, \cos\left(2 \, t\right) e^{\left(-4 \, t\right)} = -16 \, {y} - 4 \, {y'} \]

Answer:

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


Example 41 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -4 \, {y'} = -32 \, e^{t} \sin\left(-4 \, t\right) - 4 \, {y} \]

Answer:

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


Example 42 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 0 = -2 \, {y'} - 4 \, {y} + 6 \, e^{\left(-2 \, t\right)} \]

Answer:

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


Example 43 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 3 \, {y'} = 12 \, {y} - 45 \, e^{\left(-t\right)} \]

Answer:

\[ {y} = k e^{\left(4 \, t\right)} + 3 \, e^{\left(-t\right)} \]


Example 44 πŸ”—

Explain how to find the general solution to the given ODE.

\[ -2 \, {y} = 2 \, {y'} - 4 \, e^{\left(-t\right)} \]

Answer:

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


Example 45 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 48 \, e^{\left(5 \, t\right)} = -3 \, {y'} - 9 \, {y} \]

Answer:

\[ {y} = k e^{\left(-3 \, t\right)} - 2 \, e^{\left(5 \, t\right)} \]


Example 46 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 6 \, e^{\left(3 \, t\right)} = -3 \, {y'} + 9 \, {y} \]

Answer:

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


Example 47 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 0 = 36 \, e^{\left(2 \, t\right)} \sin\left(-3 \, t\right) + 8 \, {y} - 4 \, {y'} \]

Answer:

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


Example 48 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 2 \, {y'} + 2 \, {y} = 12 \, \cos\left(-2 \, t\right) e^{\left(-t\right)} \]

Answer:

\[ {y} = k e^{\left(-t\right)} - 3 \, e^{\left(-t\right)} \sin\left(-2 \, t\right) \]


Example 49 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 3 \, {y} = 10 \, e^{\left(-3 \, t\right)} \sin\left(5 \, t\right) - {y'} \]

Answer:

\[ {y} = k e^{\left(-3 \, t\right)} - 2 \, \cos\left(5 \, t\right) e^{\left(-3 \, t\right)} \]


Example 50 πŸ”—

Explain how to find the general solution to the given ODE.

\[ 4 \, e^{\left(-5 \, t\right)} = -10 \, {y} - 2 \, {y'} \]

Answer:

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