91Ó°ÊÓ

Given equation is$(D^2+9)y=30\sin 3x$

A general solution to the nth order differential equation is one that incorporates a significant number of arbitrary constants. If one uses the variable approach to solve a first-order differential equation, one must insert an arbitrary constant as soon as integration is completed.

Substitute the value as,

$D=y^{\text{'}},D^2=y^{\text{'}\text{'}}$

$(D^2+9)y=30\sin 3x$

The auxiliary equation can be written as

$\begin{array}{c}{m}^2+9=0\\ m=\sqrt{−9}\end{array}$

(Solveby the use of discriminant method)

The roots are$⇒m=Â\pm 3i$

The complementary function is

$\begin{array}{c}P.I=\frac{1}{D^2+9}30\sin 3x\\ =\frac{1}{D^2+9}30\sin 3x\end{array}$

(putting ,${\text{D}}^2=−a^2$denominator becomes 0 )

$\begin{array}{l}⇒\frac{1}{2D}30x\sin 3x\text{ }\\ ⇒\frac{−15x\cos 3x}{3}\\ ⇒−5x\cos 3x\end{array}$

(Differentiating denominator by and multiplying numeratorby x)

P.$I=−5x\cos 3x$

C.$S=C_1\cos 3x+C_2\sin 3x−5x\cos 3x$

The solution of the differential equation can be written as

$y\left(x\right)=C_1\cos 3x+C_2\sin 3x−5x\cos 3x$