91Ó°ÊÓ

Given equation is$(D^2+2D+10)=100\cos 4x$

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.

Substitute the values in given equation is

$\begin{array}{c}D=y^{\text{'}},D^2=y^{\text{'}\text{'}}\\ y^{\text{'}}+2y^{\text{'}}+10y=100\cos 4x\\ (D^2+2D+10)=10x)\cos 4x\end{array}$

The auxiliary equation can be written as

$m^2+2m+10=0$

Use discriminate method

$\begin{array}{c}m=−\frac{bÂ\pm \sqrt{b^2−4ax}}{2x}\\ m=\frac{−2Â\pm \sqrt{4−40}}{2}\\ m=\frac{−2Â\pm \sqrt{−36}}{2}\\ m=−1Â\pm 3i\end{array}$

C.F$=(C_1\sin 3x+C_2\cos 3x)e^{−s}$

$\begin{array}{l}P.l=\frac{1}{D^2+2D+10}100\cos 4x\\ \left(\begin{array}{l}{D}^2=−a^2\\ \end{array}\right)\end{array}$

Where a is 4

$\begin{array}{l}\frac{1}{−16+2D+10}100\cos 4x\\ P.I=\frac{1}{D^2+2D+10}100\cos 4x\end{array}$

$\left(\begin{array}{l}{D}^2=−a^2\\ \end{array}\right)$Where a is 4

$\begin{array}{c}\frac{1}{−16420+10}100\cos 4x\\ \frac{1}{2n−6}100\cos 4x\\ \frac{2D_{16}}{(2n−6x2n+6)}100\cos 4x\end{array}$

Now rationalize the denominator

$\begin{array}{l}\frac{2D_{16}}{4D^2−36}100\cos 4x\text{ â¶Ä‰â¶Ä‰}\\ \frac{212+6}{4[−16)−36}100\cos 4x\\ \frac{2016}{−100}100\cos 4x\end{array}$where a is 4

$\begin{array}{l}−(2D+6)\cos 4x\\ −(2D\cos 4x+6\cos 4x)\end{array}$

D here is differentiation

$\begin{array}{c}−(−8\sin 4x+6\cos 4x)\\ P.l=8\sin 4x+6\cos 4x\\ ∴C.S=C.F+P.I\end{array}$

Hence$\text{C}S=(C_1\sin 3x+C_2\cos 3x)e^{−x}+8\sin 4x+6\cos 4x$

Therefore, the solution of the differential equation is

$y\left(x\right)=(C_1\sin 3x+C_2\cos 3x)e^{−x}+8\sin 4x+6\cos 4x$