## C5 - Non-homogeneous second-order linear ODE

#### Example 1 🔗

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

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

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

#### Example 2 🔗

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

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

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

#### Example 3 🔗

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

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

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

#### Example 4 🔗

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

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

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

#### Example 5 🔗

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

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

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

#### Example 6 🔗

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

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

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

#### Example 7 🔗

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

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

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

#### Example 8 🔗

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

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

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

#### Example 9 🔗

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

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

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

#### Example 10 🔗

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

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

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

#### Example 11 🔗

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

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

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

#### Example 12 🔗

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

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

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

#### Example 13 🔗

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

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

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

#### Example 14 🔗

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

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

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

#### Example 15 🔗

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

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

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

#### Example 16 🔗

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

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

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

#### Example 17 🔗

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

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

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

#### Example 18 🔗

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

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

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

#### Example 19 🔗

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

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

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

#### Example 20 🔗

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

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

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

#### Example 21 🔗

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

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

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

#### Example 22 🔗

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

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

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

#### Example 23 🔗

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

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

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

#### Example 24 🔗

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

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

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

#### Example 25 🔗

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

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

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

#### Example 26 🔗

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

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

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

#### Example 27 🔗

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

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

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

#### Example 28 🔗

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

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

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

#### Example 29 🔗

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

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

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

#### Example 30 🔗

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

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

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

#### Example 31 🔗

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

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

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

#### Example 32 🔗

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

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

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

#### Example 33 🔗

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

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

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

#### Example 34 🔗

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

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

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

#### Example 35 🔗

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

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

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

#### Example 36 🔗

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

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

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

#### Example 37 🔗

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

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

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

#### Example 38 🔗

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

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

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

#### Example 39 🔗

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

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

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

#### Example 40 🔗

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

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

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

#### Example 41 🔗

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

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

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

#### Example 42 🔗

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

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

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

#### Example 43 🔗

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

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

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

#### Example 44 🔗

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

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

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

#### Example 45 🔗

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

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

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

#### Example 46 🔗

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

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

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

#### Example 47 🔗

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

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

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

#### Example 48 🔗

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

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

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

#### Example 49 🔗

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

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

${y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} - \sin\left(5 \, t\right)$
${y'} - {y''} + 6 \, {y} + 10 \, \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right) = 0$
${y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} - \cos\left(2 \, t\right)$