## C2 - Non-homogeneous first-order linear ODE

#### Example 1 π

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

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

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

#### Example 2 π

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

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

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

#### Example 3 π

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

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

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

#### Example 4 π

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

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

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

#### Example 5 π

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

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

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

#### Example 6 π

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

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

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

#### Example 7 π

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

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

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

#### Example 8 π

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

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

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

#### Example 9 π

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

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

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

#### Example 10 π

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

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

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

#### Example 11 π

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

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

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

#### Example 12 π

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

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

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

#### Example 13 π

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

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

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

#### Example 14 π

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

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

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

#### Example 15 π

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

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

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

#### Example 16 π

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

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

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

#### Example 17 π

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

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

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

#### Example 18 π

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

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

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

#### Example 19 π

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

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

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

#### Example 20 π

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

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

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

#### Example 21 π

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

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

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

#### Example 22 π

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

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

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

#### Example 23 π

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

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

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

#### Example 24 π

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

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

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

#### Example 25 π

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

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

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

#### Example 26 π

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

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

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

#### Example 27 π

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

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

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

#### Example 28 π

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

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

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

#### Example 29 π

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

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

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

#### Example 30 π

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

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

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

#### Example 31 π

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

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

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

#### Example 32 π

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

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

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

#### Example 33 π

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

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

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

#### Example 34 π

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

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

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

#### Example 35 π

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

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

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

#### Example 36 π

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

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

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

#### Example 37 π

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

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

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

#### Example 38 π

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

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

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

#### Example 39 π

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

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

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

#### Example 40 π

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

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

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

#### Example 41 π

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

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

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

#### Example 42 π

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

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

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

#### Example 43 π

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

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

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

#### Example 44 π

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

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

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

#### Example 45 π

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

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

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

#### Example 46 π

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

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

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

#### Example 47 π

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

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

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

#### Example 48 π

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

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

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

#### Example 49 π

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

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

${y} = k e^{\left(-3 \, t\right)} - 2 \, \cos\left(5 \, t\right) e^{\left(-3 \, t\right)}$
$4 \, e^{\left(-5 \, t\right)} = -10 \, {y} - 2 \, {y'}$
${y} = k e^{\left(-5 \, t\right)} - 2 \, t e^{\left(-5 \, t\right)}$