Question

Solve the following differential equations by the methods discussed above and compare computer solutions.

$D(D+5)y=0$

Short answer

The solution is$y=c_1+c_2{e}^{-5x}$

Step-by-step solution

Given information

A differential equation is given as $D(D+5)y=0$.

Differential equation. 

A differential equation of the form $(D-a)(D-b)y=0,a\ne b$has general solution

$y=c_1{e}^{ax}+c_2{e}^{bx}.$

Find the solution of the given differential equation. 

Suppose that $D=\frac{d}{dx}$.

Then

$$\begin{gathered} Dy=\frac{dy}{dx} \\ Dy=y^{\text{'}} \\ {D}^{2}y=\frac{d}{dx}\left(\frac{dy}{dx}\right) \\ \frac{d^{2}y}{d{x}^2}=y^{\text{'}\text{'}} \end{gathered}$$

Given differential equation is$D(D+5)y=0$.

Auxiliary equation is-

$D(D+5)=0$

These are unequal roots$\Rightarrow D=0,D=-5$

Differential equation becomes

$\Rightarrow (D-0)[D-(-5\left)\right]y=0$

These are separable equation with solution

$y=c_1{e}^{0x}\text{and}y=c_2{e}^{-5x}$

The general solution will be given by

$$\begin{gathered} y=c_1{e}^{0x}+c_2{e}^{-5x} \\ y=c_1+c_2{e}^{-5x} \end{gathered}$$