## C4 - Homogeneous second-order linear IVP

#### Example 1 🔗

Explain how to find the particular solution to each given IVP.

1. $-3 \, {x'} - {x''} = -10 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -16$

2. $81 \, {y} = -{y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= -9$

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

${y} = \cos\left(9 \, t\right) - \sin\left(9 \, t\right)$

#### Example 2 🔗

Explain how to find the particular solution to each given IVP.

1. $0 = 100 \, {x} + {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -50$

2. $-{y''} - 8 \, {y'} = 15 \, {y} \hspace{2em} {y} (0)= -2 , {y} '(0)= 10$

${x} = 3 \, \cos\left(10 \, t\right) - 5 \, \sin\left(10 \, t\right)$

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

#### Example 3 🔗

Explain how to find the particular solution to each given IVP.

1. $16 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -20$

2. ${y''} = 25 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -50$

${x} = 2 \, \cos\left(4 \, t\right) - 5 \, \sin\left(4 \, t\right)$

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

#### Example 4 🔗

Explain how to find the particular solution to each given IVP.

1. ${x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 4$

2. $4 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 4$

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

${y} = 2 \, \sin\left(2 \, t\right)$

#### Example 5 🔗

Explain how to find the particular solution to each given IVP.

1. $-9 \, {x} = {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -9$

2. $0 = {y''} - 9 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= 12$

${x} = 2 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)$

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

#### Example 6 🔗

Explain how to find the particular solution to each given IVP.

1. $-2 \, {x} + {x'} = -{x''} \hspace{2em} {x} (0)= 7 , {x} '(0)= -8$

2. $0 = {y''} + 25 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -15$

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

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

#### Example 7 🔗

Explain how to find the particular solution to each given IVP.

1. $0 = {x''} + 64 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 24$

2. ${y'} + 2 \, {y} = {y''} \hspace{2em} {y} (0)= 7 , {y} '(0)= -1$

${x} = \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)$

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

#### Example 8 🔗

Explain how to find the particular solution to each given IVP.

1. $0 = 6 \, {x} + {x''} + 5 \, {x'} \hspace{2em} {x} (0)= -4 , {x} '(0)= 11$

2. $0 = -25 \, {y} - {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -25$

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

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

#### Example 9 🔗

Explain how to find the particular solution to each given IVP.

1. $-16 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 0$

2. $12 \, {y} = {y'} + {y''} \hspace{2em} {y} (0)= 9 , {y} '(0)= -1$

${x} = -3 \, \cos\left(4 \, t\right)$

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

#### Example 10 🔗

Explain how to find the particular solution to each given IVP.

1. ${x''} + 16 \, {x} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -16$

2. $-3 \, {y'} = -2 \, {y} - {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= 0$

${x} = -5 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)$

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

#### Example 11 🔗

Explain how to find the particular solution to each given IVP.

1. $-100 \, {x} = {x''} \hspace{2em} {x} (0)= 0 , {x} '(0)= -10$

2. $-5 \, {y} = -{y''} + 4 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 19$

${x} = -\sin\left(10 \, t\right)$

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

#### Example 12 🔗

Explain how to find the particular solution to each given IVP.

1. $9 \, {x'} - {x''} = 20 \, {x} \hspace{2em} {x} (0)= -2 , {x} '(0)= -6$

2. $-{y''} = 25 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= -25$

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

${y} = -4 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)$

#### Example 13 🔗

Explain how to find the particular solution to each given IVP.

1. ${x''} - 8 \, {x} = -2 \, {x'} \hspace{2em} {x} (0)= 5 , {x} '(0)= -8$

2. ${y''} = -9 \, {y} \hspace{2em} {y} (0)= -2 , {y} '(0)= -15$

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

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

#### Example 14 🔗

Explain how to find the particular solution to each given IVP.

1. $0 = -{x''} - 4 \, {x} \hspace{2em} {x} (0)= 5 , {x} '(0)= 2$

2. $-{y''} - 6 \, {y} = -5 \, {y'} \hspace{2em} {y} (0)= 6 , {y} '(0)= 16$

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

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

#### Example 15 🔗

Explain how to find the particular solution to each given IVP.

1. $15 \, {x} = 2 \, {x'} + {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 31$

2. $0 = -64 \, {y} - {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= 0$

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

${y} = 5 \, \cos\left(8 \, t\right)$

#### Example 16 🔗

Explain how to find the particular solution to each given IVP.

1. ${x''} = -16 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -8$

2. $-{y'} = {y''} - 12 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -14$

${x} = -3 \, \cos\left(4 \, t\right) - 2 \, \sin\left(4 \, t\right)$

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

#### Example 17 🔗

Explain how to find the particular solution to each given IVP.

1. ${x''} = 5 \, {x} + 4 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 3$

2. ${y''} + 49 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= -21$

${x} = -3 \, e^{\left(-t\right)}$

${y} = 4 \, \cos\left(7 \, t\right) - 3 \, \sin\left(7 \, t\right)$

#### Example 18 🔗

Explain how to find the particular solution to each given IVP.

1. $-15 \, {x} - {x''} = -8 \, {x'} \hspace{2em} {x} (0)= -4 , {x} '(0)= -10$

2. $-25 \, {y} = {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -10$

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

${y} = 5 \, \cos\left(5 \, t\right) - 2 \, \sin\left(5 \, t\right)$

#### Example 19 🔗

Explain how to find the particular solution to each given IVP.

1. $0 = 4 \, {x} - 3 \, {x'} - {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -8$

2. $0 = {y''} + 25 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 0$

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

${y} = \cos\left(5 \, t\right)$

#### Example 20 🔗

Explain how to find the particular solution to each given IVP.

1. $-7 \, {x'} + 12 \, {x} = -{x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10$

2. $4 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= 10$

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

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

#### Example 21 🔗

Explain how to find the particular solution to each given IVP.

1. $0 = -12 \, {x} + {x''} + {x'} \hspace{2em} {x} (0)= 1 , {x} '(0)= 3$

2. $0 = -{y''} - 36 \, {y} \hspace{2em} {y} (0)= -2 , {y} '(0)= -24$

${x} = e^{\left(3 \, t\right)}$

${y} = -2 \, \cos\left(6 \, t\right) - 4 \, \sin\left(6 \, t\right)$

#### Example 22 🔗

Explain how to find the particular solution to each given IVP.

1. $0 = 9 \, {x} + {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -9$

2. $-8 \, {y} - {y''} = -6 \, {y'} \hspace{2em} {y} (0)= -7 , {y} '(0)= -24$

${x} = -4 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)$

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

#### Example 23 🔗

Explain how to find the particular solution to each given IVP.

1. $7 \, {x'} + 10 \, {x} = -{x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -18$

2. $-25 \, {y} = {y''} \hspace{2em} {y} (0)= 4 , {y} '(0)= -5$

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

${y} = 4 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)$

#### Example 24 🔗

Explain how to find the particular solution to each given IVP.

1. ${x''} = 4 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -18$

2. ${y''} = -16 \, {y} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12$

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

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

#### Example 25 🔗

Explain how to find the particular solution to each given IVP.

1. ${x} = {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= 4$

2. $0 = -36 \, {y} - {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= 18$

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

${y} = -2 \, \cos\left(6 \, t\right) + 3 \, \sin\left(6 \, t\right)$

#### Example 26 🔗

Explain how to find the particular solution to each given IVP.

1. $0 = 25 \, {x} + {x''} \hspace{2em} {x} (0)= 0 , {x} '(0)= 5$

2. $-{y''} + 5 \, {y} = 4 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 29$

${x} = \sin\left(5 \, t\right)$

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

#### Example 27 🔗

Explain how to find the particular solution to each given IVP.

1. $36 \, {x} = -{x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -6$

2. $-3 \, {y} - {y''} = -4 \, {y'} \hspace{2em} {y} (0)= 0 , {y} '(0)= 8$

${x} = -4 \, \cos\left(6 \, t\right) - \sin\left(6 \, t\right)$

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

#### Example 28 🔗

Explain how to find the particular solution to each given IVP.

1. $15 \, {x} = -8 \, {x'} - {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= -14$

2. $0 = {y''} + 9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3$

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

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

#### Example 29 🔗

Explain how to find the particular solution to each given IVP.

1. $-8 \, {x} = 2 \, {x'} - {x''} \hspace{2em} {x} (0)= 6 , {x} '(0)= 0$

2. $0 = -9 \, {y} - {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= -9$

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

${y} = \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)$

#### Example 30 🔗

Explain how to find the particular solution to each given IVP.

1. ${x''} + 9 \, {x} = 0 \hspace{2em} {x} (0)= -1 , {x} '(0)= 3$

2. $7 \, {y'} = -{y''} - 12 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -1$

${x} = -\cos\left(3 \, t\right) + \sin\left(3 \, t\right)$

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

#### Example 31 🔗

Explain how to find the particular solution to each given IVP.

1. ${x''} = -25 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10$

2. ${y''} - {y} = 0 \hspace{2em} {y} (0)= 1 , {y} '(0)= -9$

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

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

#### Example 32 🔗

Explain how to find the particular solution to each given IVP.

1. $-5 \, {x'} + 4 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 5 , {x} '(0)= 14$

2. $64 \, {y} = -{y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -8$

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

${y} = 3 \, \cos\left(8 \, t\right) - \sin\left(8 \, t\right)$

#### Example 33 🔗

Explain how to find the particular solution to each given IVP.

1. ${x''} + 16 \, {x} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= 12$

2. $-{y'} - {y''} = -20 \, {y} \hspace{2em} {y} (0)= 9 , {y} '(0)= 0$

${x} = -5 \, \cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)$

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

#### Example 34 🔗

Explain how to find the particular solution to each given IVP.

1. ${x''} + 9 \, {x} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= -3$

2. $-{y'} - {y''} = -12 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -6$

${x} = -3 \, \cos\left(3 \, t\right) - \sin\left(3 \, t\right)$

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

#### Example 35 🔗

Explain how to find the particular solution to each given IVP.

1. $-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -2 , {x} '(0)= -8$

2. ${y'} = -{y''} + 20 \, {y} \hspace{2em} {y} (0)= -6 , {y} '(0)= -15$

${x} = -2 \, \cos\left(2 \, t\right) - 4 \, \sin\left(2 \, t\right)$

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

#### Example 36 🔗

Explain how to find the particular solution to each given IVP.

1. ${x''} - 9 \, {x} = 0 \hspace{2em} {x} (0)= 3 , {x} '(0)= 9$

2. $-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12$

${x} = 3 \, e^{\left(3 \, t\right)}$

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

#### Example 37 🔗

Explain how to find the particular solution to each given IVP.

1. $6 \, {x} + {x''} - 5 \, {x'} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= -7$

2. $4 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 4$

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

${y} = 2 \, \sin\left(2 \, t\right)$

#### Example 38 🔗

Explain how to find the particular solution to each given IVP.

1. $0 = 4 \, {x} + {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= -10$

2. ${y''} = -2 \, {y'} + 15 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= 2$

${x} = -3 \, \cos\left(2 \, t\right) - 5 \, \sin\left(2 \, t\right)$

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

#### Example 39 🔗

Explain how to find the particular solution to each given IVP.

1. $0 = -64 \, {x} - {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -8$

2. $-5 \, {y'} + {y''} = -4 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 7$

${x} = -4 \, \cos\left(8 \, t\right) - \sin\left(8 \, t\right)$

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

#### Example 40 🔗

Explain how to find the particular solution to each given IVP.

1. $-5 \, {x} + 4 \, {x'} = -{x''} \hspace{2em} {x} (0)= -6 , {x} '(0)= 24$

2. ${y''} + 9 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -6$

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

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

#### Example 41 🔗

Explain how to find the particular solution to each given IVP.

1. $-{x''} = 36 \, {x} \hspace{2em} {x} (0)= -5 , {x} '(0)= 12$

2. $-4 \, {y'} = 3 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 7$

${x} = -5 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)$

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

#### Example 42 🔗

Explain how to find the particular solution to each given IVP.

1. $-12 \, {x} = {x''} + 7 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 7$

2. ${y''} = -4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= -10$

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

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

#### Example 43 🔗

Explain how to find the particular solution to each given IVP.

1. $-{x''} - 3 \, {x} = -4 \, {x'} \hspace{2em} {x} (0)= 7 , {x} '(0)= 11$

2. $-{y''} = 100 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= -10$

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

${y} = \cos\left(10 \, t\right) - \sin\left(10 \, t\right)$

#### Example 44 🔗

Explain how to find the particular solution to each given IVP.

1. $-2 \, {x'} - 3 \, {x} = -{x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= -15$

2. $36 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= 6$

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

${y} = -\cos\left(6 \, t\right) + \sin\left(6 \, t\right)$

#### Example 45 🔗

Explain how to find the particular solution to each given IVP.

1. ${x''} = 5 \, {x'} - 4 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= 5$

2. $0 = -9 \, {y} - {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -9$

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

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

#### Example 46 🔗

Explain how to find the particular solution to each given IVP.

1. $-3 \, {x'} + {x''} = -2 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= 3$

2. ${y''} + 36 \, {y} = 0 \hspace{2em} {y} (0)= -4 , {y} '(0)= 18$

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

${y} = -4 \, \cos\left(6 \, t\right) + 3 \, \sin\left(6 \, t\right)$

#### Example 47 🔗

Explain how to find the particular solution to each given IVP.

1. ${x''} + 81 \, {x} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 9$

2. $6 \, {y} = -5 \, {y'} - {y''} \hspace{2em} {y} (0)= 7 , {y} '(0)= -18$

${x} = -3 \, \cos\left(9 \, t\right) + \sin\left(9 \, t\right)$

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

#### Example 48 🔗

Explain how to find the particular solution to each given IVP.

1. $-16 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -4 , {x} '(0)= 8$

2. $-20 \, {y} = {y'} - {y''} \hspace{2em} {y} (0)= 4 , {y} '(0)= -7$

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

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

#### Example 49 🔗

Explain how to find the particular solution to each given IVP.

1. $-{x''} - 9 \, {x} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= 6$

2. $-{y''} = -2 \, {y'} - 3 \, {y} \hspace{2em} {y} (0)= -6 , {y} '(0)= -2$

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

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

#### Example 50 🔗

Explain how to find the particular solution to each given IVP.

1. $-{x''} = 16 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= 8$

2. $-4 \, {y'} + 5 \, {y} = {y''} \hspace{2em} {y} (0)= 9 , {y} '(0)= -15$

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

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