Question

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

$\mathbf{3}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{10}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{15}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

The general solution is$y\left(x\right)=c_1{e}^{-\frac{1}{3}x}+e^{-\frac{3}{2}}[c_2\cos \sqrt{7}x+c_3\sin \sqrt{7}x]$.

Step-by-step solution

Given Data

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

Given differential equation,

$3y\text{'}\text{'}\text{'}+10y\text{'}\text{'}+15y\text{'}+4y=0$

Find auxiliary equation

The auxiliary equation is$3m^3+10m^2+15m+4=0$

If$m=-\frac{1}{3}$, then$3m^3+10m^2+15m+4=0$

Therefore, $m=-\frac{1}{3}$is a root.

So,

$(3m+1)(3m^2+9m+12)=0$

Find general solution

$$\begin{gathered} (3m+1)(3m^2+9m+12)=0 \\ (3m+1)(m^2+3m+4)=0 \\ \therefore m=-\frac{1}{3},m=\frac{-3\pm \sqrt{7}i}{2} \end{gathered}$$

and

$$\begin{gathered} m^2+3m+4=0 \\ m=\frac{-3\pm \sqrt{9-16}}{2} \\ m=\frac{-3\pm \sqrt{7}i}{2} \end{gathered}$$

$y\left(x\right)=c_1{e}^{-\frac{1}{3}x}+e^{-\frac{3}{2}}[c_2\cos \sqrt{7}x+c_3\sin \sqrt{7}x]$