C4 - Homogeneous second-order linear IVP


Example 1 πŸ”—

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

Answer:

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

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


Example 2 πŸ”—

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

Answer:

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

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


Example 3 πŸ”—

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

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

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

Answer:

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

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


Example 4 πŸ”—

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

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

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

Answer:

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

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


Example 5 πŸ”—

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

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

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

Answer:

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

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


Example 6 πŸ”—

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

Answer:

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

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


Example 7 πŸ”—

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

Answer:

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

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


Example 8 πŸ”—

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

Answer:

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

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


Example 9 πŸ”—

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

Answer:

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

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


Example 10 πŸ”—

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

Answer:

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

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


Example 11 πŸ”—

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

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

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

Answer:

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

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


Example 12 πŸ”—

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

Answer:

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

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


Example 13 πŸ”—

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

Answer:

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

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


Example 14 πŸ”—

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

Answer:

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

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


Example 15 πŸ”—

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

Answer:

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

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


Example 16 πŸ”—

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

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

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

Answer:

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

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


Example 17 πŸ”—

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

Answer:

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

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


Example 18 πŸ”—

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

Answer:

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

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


Example 19 πŸ”—

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

Answer:

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

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


Example 20 πŸ”—

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

Answer:

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

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


Example 21 πŸ”—

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

Answer:

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

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


Example 22 πŸ”—

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

Answer:

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

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


Example 23 πŸ”—

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

Answer:

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

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


Example 24 πŸ”—

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

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

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

Answer:

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

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


Example 25 πŸ”—

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

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

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

Answer:

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

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


Example 26 πŸ”—

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

Answer:

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

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


Example 27 πŸ”—

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

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

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

Answer:

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

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


Example 28 πŸ”—

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

Answer:

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

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


Example 29 πŸ”—

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

Answer:

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

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


Example 30 πŸ”—

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

Answer:

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

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


Example 31 πŸ”—

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

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

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

Answer:

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

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


Example 32 πŸ”—

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

Answer:

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

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


Example 33 πŸ”—

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

Answer:

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

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


Example 34 πŸ”—

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

Answer:

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

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


Example 35 πŸ”—

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

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

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

Answer:

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

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


Example 36 πŸ”—

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

Answer:

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

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


Example 37 πŸ”—

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

Answer:

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

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


Example 38 πŸ”—

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

Answer:

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

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


Example 39 πŸ”—

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

Answer:

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

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


Example 40 πŸ”—

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

Answer:

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

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


Example 41 πŸ”—

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

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

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

Answer:

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

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


Example 42 πŸ”—

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

Answer:

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

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


Example 43 πŸ”—

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

Answer:

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

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


Example 44 πŸ”—

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

Answer:

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

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


Example 45 πŸ”—

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

Answer:

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

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


Example 46 πŸ”—

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

Answer:

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

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


Example 47 πŸ”—

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

Answer:

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

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


Example 48 πŸ”—

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

Answer:

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

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


Example 49 πŸ”—

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

Answer:

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

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


Example 50 πŸ”—

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

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

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

Answer:

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

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