Question

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

$\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{12}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{.}$

Short answer

The general solution of each differential equation is

$y_c=\left(c_1+c_{2}x\right)e^x+c_3{e}^{-\frac{5}{2}x}$

Step-by-step solution

Given Data

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

Given differential equation,

$2y\text{'}\text{'}+9y\text{'}\text{'}+12y\text{'}\text{'}+5y=0.$

In operator form,

$[2D^3+9D^2+12D+5]y=0$

Write the characteristics equation

Characteristic equation,

$2m^3+9m^2+12m+5=0$

by substitution method:

$$\begin{gathered} Form=-1\Rightarrow -2+9-12+5 \\ =-14+14 \\ =0 \end{gathered}$$

Final calculation

Simplify,

$$\begin{gathered} (m+1)(2m^2+7m+5)=0 \\ (m+1)(2m^2+2m+5m+5)=0 \\ (m+1)\left(2m\right(m+1)+5(m+1\left)\right)=0 \\ (m+1)\left(\right(m+1\left)\right(2m+5\left)\right)=0 \\ \Rightarrow {(m+1)}^2(2m+5)=0 \\ m=-1,-1,-\frac{5}{2} \end{gathered}$$

Hence, the general solution of each differential equation is

$y_c=\left(c_1+c_{2}x\right)e^{-x}+c_3{e}^{-\frac{5}{2}x}$