Question

Find the general solution of the following differential equations (complementary function + particular solution). Find the particular solution by inspection. Also find a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form.

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{8}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{25}\mathit{y}\mathbf{=}\mathbf{120}\mathit{s}\mathit{i}\mathit{n}\mathbf{5}\mathit{x}$

Short answer

The solution of differential equation is$y\left(x\right)=(C_1\sin 3x+C_2\cos 3x)e^{-4x}-3\cos 5x$

Step-by-step solution

Given information

Given equation is$y^{\text{'}\text{'}}+8y^{\text{'}}+25y=120\sin 5x$

Definition of differential equation 

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself to its derivatives of various orders.

Solve the given differential equation 

Substitute in given equation is

$\begin{array}{c}D=y\text{'}\\ D^2=y"\\ y^N+8y^{\text{'}}+25y=120\sin 5x\\ (D^2+8D+25)y=120\sin 5x\end{array}$

The auxiliary equation can be written as

Use Discriminant Method,

$\begin{array}{c}{m}^2+8m+25=0.\\ m=\frac{-x_1\sqrt{\omega -1m}}{2}\\ =\frac{-1-4}{2}\\ =-4\pm 3i\end{array}$

The values of C.F & P.I

The values of C.F & P.I is given as

$\begin{array}{c}=(C_1\sin 3x+C_2\cos 3x)e^{-4x}\\ =\frac{1}{5^2+2x+23}120\sin 5x\end{array}$

Solve Further the equation

$\begin{array}{c}\frac{1}{3d+25}120\sin 5x\\ \frac{1}{25d}120\sin 5x\\ -\frac{1}{\pi }120\cos 5x\\ -3\cos 5x\end{array}$

(Put $D^2=-a^2$where a here is 5)

($\pi =$integration and D is differentiation)

P.I$=-3\cos 5x$

Hence C.S$=(C_1\sin 3x+C_2\cos 3x)e^{-4x}-3\cos 5x$

The solution of the differential equation can be written as

$y\left(x\right)=(C_1\sin 3x+C_2\cos 3x)e^{-4x}-3\cos 5x$