91Ó°ÊÓ

Given differential equation is$(D-1)(D+3)(D+5)y=0$

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

$y=c_1{e}^{ax}+c_2{e}^{bx}+c_3{e}^{cx}$ if a, b, c is all different and real.

A differential equation is given as$(D-1)(D+3)(D+5)y=0$.

Let us suppose$D=\frac{d}{dx}$.

Then

$$\begin{gathered} dDy=\frac{dy}{dx} \\ =y^{\text{'}} \end{gathered}$$

The second order differential equation

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

The third order differential equation

$$\begin{gathered} D^{3}y=\frac{d}{dx}\left(\frac{d^{2}y}{d{x}^2}\right) \\ =\frac{d^{3}y}{d{x}^3} \\ =y^{\text{'}\text{'}\text{'}} \end{gathered}$$

Given differential equation is $(D-1)(D+3)(D+5)y=0$.

Compare with standard third order differential equation of the form

$(D-a)(D-b)(D-c)y=0$

$a=1,b=-3,c=-5.$

Therefore, general solution will be given as:

$$\begin{gathered} y=c_1{e}^{ax}+c_2{e}^{bx}+c_3{e}^{cx} \\ y=c_1{e}^{1x}+c_2{e}^{-3x}+c_3{e}^{-5x} \\ y=c_1{e}^x+c_2{e}^{-3x}+c_3{e}^{-5x} \end{gathered}$$

Hence, the given third order differential equation has a general solution

$y=c_1{e}^x+c_2{e}^{-3x}+c_3{e}^{-5x}.$