Question

In Problems 17-32 use the procedures developed in this chapter to find the general solution of each differential equation.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{10}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{25}\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathbf{0}$

Short answer

The general solution of each differential equation is

$y=c_1{e}^{0x^{}}+\left(c_2+c_{3}x\right)e^{-5x}=c_1+\left(c_2+c_{3}x\right)e^{-5x}$

Step-by-step solution

Given Data

Differential equation is the equation which relate one or more unknown functions and their derivatives.

The given differential equation is,

$$\begin{gathered} y\text{'}\text{'}\text{'}+10y"+25y\text{'}=0. \\ Inoperatorform, \\ \left(D^3+10D^2+25D\right)y=0 \end{gathered}$$

Write the characteristics equation

Characteristic equation,

$$\begin{gathered} D^3+10D^2+25D=0 \\ D\left(D^2+10D+25\right)=0 \\ D{\left(D+5\right)}^2=0 \\ D\left(D+5\right)\left(D+5\right)=0 \\ D=0,-5,-5. \end{gathered}$$

Hence the required general solution is,

$y=c_1{e}^{0x}+\left(c_2+c_{3}x\right)e^{-5x}=c_1+\left(c_2+c_{3}x\right)e^{-5x},$

$c_1,c_2,c_3$are arbitrary constants