C5 - Non-homogeneous second-order linear ODE


Example 1 πŸ”—

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

\[ 60 \, t e^{\left(3 \, t\right)} + 6 \, {y} + 22 \, e^{\left(3 \, t\right)} = -{y''} - 5 \, {y'} \]

Answer:

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


Example 2 πŸ”—

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

\[ 6 \, t e^{\left(2 \, t\right)} - 12 \, {y} - 9 \, e^{\left(2 \, t\right)} = -7 \, {y'} + {y''} \]

Answer:

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


Example 3 πŸ”—

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

\[ 0 = 48 \, t e^{\left(5 \, t\right)} + {y''} - 9 \, {y} + 30 \, e^{\left(5 \, t\right)} \]

Answer:

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


Example 4 πŸ”—

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

\[ -{y''} + 3 \, {y'} - 2 \, {y} = -9 \, \cos\left(3 \, t\right) - 7 \, \sin\left(3 \, t\right) \]

Answer:

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


Example 5 πŸ”—

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

\[ 5 \, {y'} - {y''} = 4 \, {y} - 20 \, \cos\left(2 \, t\right) \]

Answer:

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


Example 6 πŸ”—

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

\[ 8 \, {y} = -35 \, t e^{\left(3 \, t\right)} - {y''} - 6 \, {y'} - 12 \, e^{\left(3 \, t\right)} \]

Answer:

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


Example 7 πŸ”—

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

\[ 3 \, {y} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = {y''} - 2 \, {y'} \]

Answer:

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


Example 8 πŸ”—

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

\[ 0 = 12 \, t e^{\left(3 \, t\right)} - 2 \, {y} + {y''} - {y'} + 15 \, e^{\left(3 \, t\right)} \]

Answer:

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


Example 9 πŸ”—

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

\[ 0 = -12 \, {y} + {y''} + {y'} + 32 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) \]

Answer:

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


Example 10 πŸ”—

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

\[ 0 = -{y''} + 12 \, {y} - {y'} - 6 \, \cos\left(2 \, t\right) + 48 \, \sin\left(2 \, t\right) \]

Answer:

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


Example 11 πŸ”—

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

\[ -30 \, t e^{\left(3 \, t\right)} - 6 \, {y} - 11 \, e^{\left(3 \, t\right)} = 5 \, {y'} + {y''} \]

Answer:

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


Example 12 πŸ”—

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

\[ {y} = {y''} + 5 \, \cos\left(2 \, t\right) \]

Answer:

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


Example 13 πŸ”—

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

\[ 0 = 56 \, t e^{\left(5 \, t\right)} + {y''} - 2 \, {y} + {y'} + 22 \, e^{\left(5 \, t\right)} \]

Answer:

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


Example 14 πŸ”—

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

\[ {y} = {y''} - 26 \, \cos\left(5 \, t\right) \]

Answer:

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


Example 15 πŸ”—

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

\[ {y''} - 7 \, {y'} + 12 \, {y} = -70 \, \cos\left(5 \, t\right) - 26 \, \sin\left(5 \, t\right) \]

Answer:

\[ {y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 2 \, \sin\left(5 \, t\right) \]


Example 16 πŸ”—

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

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

Answer:

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


Example 17 πŸ”—

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

\[ 2 \, {y} = -{y''} - 3 \, {y'} - 18 \, \cos\left(2 \, t\right) + 6 \, \sin\left(2 \, t\right) \]

Answer:

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


Example 18 πŸ”—

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

\[ {y''} = 21 \, t e^{\left(5 \, t\right)} + 4 \, {y} + 10 \, e^{\left(5 \, t\right)} \]

Answer:

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


Example 19 πŸ”—

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

\[ {y''} - {y} - 15 \, \sin\left(2 \, t\right) = 0 \]

Answer:

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


Example 20 πŸ”—

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

\[ 5 \, {y'} + {y''} = -4 \, {y} + 30 \, \cos\left(2 \, t\right) \]

Answer:

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


Example 21 πŸ”—

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

\[ 0 = -36 \, t e^{\left(3 \, t\right)} - 3 \, {y} + {y''} + 2 \, {y'} - 24 \, e^{\left(3 \, t\right)} \]

Answer:

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


Example 22 πŸ”—

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

\[ 12 \, t e^{\left(5 \, t\right)} + 3 \, {y'} - {y''} + 14 \, e^{\left(5 \, t\right)} = -4 \, {y} \]

Answer:

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


Example 23 πŸ”—

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

\[ -4 \, {y} + 45 \, \cos\left(3 \, t\right) - 15 \, \sin\left(3 \, t\right) = 5 \, {y'} + {y''} \]

Answer:

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


Example 24 πŸ”—

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

\[ 4 \, {y} - 3 \, {y'} - {y''} = 58 \, \cos\left(5 \, t\right) + 30 \, \sin\left(5 \, t\right) \]

Answer:

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


Example 25 πŸ”—

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

\[ -2 \, {y} = -18 \, t e^{\left(5 \, t\right)} + {y'} - {y''} - 9 \, e^{\left(5 \, t\right)} \]

Answer:

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


Example 26 πŸ”—

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

\[ 0 = -4 \, {y'} + {y''} + 3 \, {y} - 44 \, \cos\left(5 \, t\right) + 40 \, \sin\left(5 \, t\right) \]

Answer:

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


Example 27 πŸ”—

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

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

Answer:

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


Example 28 πŸ”—

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

\[ 12 \, t e^{\left(5 \, t\right)} + 3 \, {y} + 8 \, e^{\left(5 \, t\right)} = {y''} - 2 \, {y'} \]

Answer:

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


Example 29 πŸ”—

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

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

Answer:

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


Example 30 πŸ”—

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

\[ 2 \, {y} = 3 \, {y'} - {y''} - 14 \, \cos\left(3 \, t\right) + 18 \, \sin\left(3 \, t\right) \]

Answer:

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


Example 31 πŸ”—

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

\[ 36 \, t e^{\left(5 \, t\right)} - 12 \, {y} + 22 \, e^{\left(5 \, t\right)} = -{y''} - {y'} \]

Answer:

\[ {y} = -2 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} \]


Example 32 πŸ”—

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

\[ -42 \, t e^{\left(3 \, t\right)} - 13 \, e^{\left(3 \, t\right)} = -7 \, {y'} - {y''} - 12 \, {y} \]

Answer:

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


Example 33 πŸ”—

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

\[ -5 \, {y'} + {y''} + 4 \, {y} = 4 \, t e^{\left(3 \, t\right)} - 2 \, e^{\left(3 \, t\right)} \]

Answer:

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


Example 34 πŸ”—

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

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

Answer:

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


Example 35 πŸ”—

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

\[ -12 \, {y} = {y'} - {y''} + 6 \, \cos\left(3 \, t\right) + 42 \, \sin\left(3 \, t\right) \]

Answer:

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


Example 36 πŸ”—

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

\[ 16 \, t e^{\left(5 \, t\right)} + 10 \, e^{\left(5 \, t\right)} = {y''} - 9 \, {y} \]

Answer:

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


Example 37 πŸ”—

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

\[ 4 \, {y} = 18 \, t e^{\left(5 \, t\right)} + {y''} - 3 \, {y'} + 21 \, e^{\left(5 \, t\right)} \]

Answer:

\[ {y} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} \]


Example 38 πŸ”—

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

\[ {y''} + 4 \, {y'} + 24 \, \cos\left(3 \, t\right) - 12 \, \sin\left(3 \, t\right) = -3 \, {y} \]

Answer:

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


Example 39 πŸ”—

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

\[ 6 \, {y} + 5 \, {y'} + {y''} = -75 \, \cos\left(5 \, t\right) + 57 \, \sin\left(5 \, t\right) \]

Answer:

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


Example 40 πŸ”—

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

\[ {y'} + 2 \, {y} - {y''} + 33 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right) = 0 \]

Answer:

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


Example 41 πŸ”—

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

\[ -12 \, {y} - 9 \, \cos\left(3 \, t\right) - 63 \, \sin\left(3 \, t\right) = -{y''} + {y'} \]

Answer:

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


Example 42 πŸ”—

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

\[ {y''} - 12 \, {y} - {y'} = -30 \, t e^{\left(2 \, t\right)} + 9 \, e^{\left(2 \, t\right)} \]

Answer:

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


Example 43 πŸ”—

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

\[ -t e^{\left(3 \, t\right)} + 8 \, {y} = 6 \, {y'} - {y''} \]

Answer:

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


Example 44 πŸ”—

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

\[ -4 \, {y} - 5 \, {y'} - {y''} + 30 \, \cos\left(3 \, t\right) - 10 \, \sin\left(3 \, t\right) = 0 \]

Answer:

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


Example 45 πŸ”—

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

\[ 5 \, {y'} + 4 \, {y} + {y''} = -25 \, \cos\left(5 \, t\right) + 21 \, \sin\left(5 \, t\right) \]

Answer:

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


Example 46 πŸ”—

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

\[ {y''} + 5 \, {y'} = -30 \, t e^{\left(3 \, t\right)} - 6 \, {y} - 11 \, e^{\left(3 \, t\right)} \]

Answer:

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


Example 47 πŸ”—

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

\[ -6 \, {y} = {y'} - {y''} - 62 \, \cos\left(5 \, t\right) + 10 \, \sin\left(5 \, t\right) \]

Answer:

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


Example 48 πŸ”—

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

\[ -{y'} + 6 \, {y} - {y''} + 6 \, \cos\left(3 \, t\right) - 30 \, \sin\left(3 \, t\right) = 0 \]

Answer:

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


Example 49 πŸ”—

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

\[ -{y''} + 7 \, {y'} = 12 \, {y} - 35 \, \cos\left(5 \, t\right) - 13 \, \sin\left(5 \, t\right) \]

Answer:

\[ {y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} - \sin\left(5 \, t\right) \]


Example 50 πŸ”—

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

\[ {y'} - {y''} + 6 \, {y} + 10 \, \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right) = 0 \]

Answer:

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