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.

${\mathbf{(}\mathbf{D}\mathbf{-}\mathbf{3}\mathbf{)}}^{\mathbf{2}}\mathit{y}\mathbf{=}\mathbf{6}{\mathit{e}}^{\mathbf{3}\mathbf{x}}$

Short answer

The solution of differential equation is$y\left(x\right)=c_1{e}^{3x}+cz{e}^{-x}+2e^{-3x}$

Step-by-step solution

Given information 

Given equation is${(D-3)}^{2}y=6e^{3x}$

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 these values in given equation is

$\begin{array}{c}D=y,D^2=y^{\text{'}\text{'}}\\ {\left(D-3\right)}^{2}y=6e^{3x}\end{array}$

The auxiliary equation can be written as

$\begin{array}{c}{m}^2-6m+9=0\\ m(m-3)-3(m-3)=0\end{array}$

The roots can be written as below

$\Rightarrow m=3,3$

The complementary function can be written as

$\begin{array}{c}\text{C.}F=(C_{1}x+C_2)e^{3x}\\ \text{P.}I=\frac{1}{0^2-5-9}6e^{3x}\\ =\frac{1}{2m}6e^{3x}\\ =\frac{1}{2}6e^{3x}\end{array}$

$\begin{array}{c}\text{P.}I=3e^{3x}\\ C.S=C.F+P.I\end{array}$

Put (D=power of the exponential only consonant term)

$C.S=(C_{1}x+C_2)e^{3x}+3e^{3x}$

(Differentiate denominator by D and multiply numerator with x )

The Solution of the differential equation can be written as

$y\left(x\right)=c{e}^{3x}+cz{e}^{-x}+2e^{-3x}.$