Question

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

$\mathbf{(}{\mathbf{D}}^{\mathbf{2}}\mathbf{+}\mathbf{9}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{30}\mathit{s}\mathit{i}\mathit{n}\mathbf{3}\mathit{x}$

Short answer

The general solution given by differential equation is

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

Step-by-step solution

Given data.

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

General solution of differential equation.

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.

Find the general solution of given differential equation.(D2+9)y=30sin3x

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$\Rightarrow m=\pm 3i$

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 )

$\begin{array}{l}\Rightarrow \frac{1}{2D}30x\sin 3x\\ \Rightarrow \frac{-15x\cos 3x}{3}\\ \Rightarrow -5x\cos 3x\end{array}$

(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

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