Question

In Problems,1-14find the general solution of the given second-order differential equation.

$\mathbf{3}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

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

Step-by-step solution

Define Second order differential equation

For a second-order differential equation $a{y}^{\text{'}\text{'}}+b{y}^{\text{'}}+cy=0$, where aandb are constants, the auxiliary equation will be of the form$a{m}^2+bm+c=0$.

There will be three forms of the general solution corresponding to three cases that depend on the solution of an auxiliary equation.

1. m₁ and m₂ are real and distinct roots that are possible when $\left(b^2-4ac>0\right)$

2. m₁ and m₂ are real and equal roots that are possible when $\left(b^2-4ac=0\right)$

3. m₁ and m₂ are conjugate complex roots $\left(b^2-4ac<0\right)$

Case I: Distinct Real Roots

The general solution of $ay\text{'}\text{'}+by\text{'}+cy=0$ on the interval $(-\infty ,\infty )$will be $y=c_1{e}^{m_{1}x}+c_2{e}^{m_{2}x}$.

Case II: Repeated Real Roots

The general solution will be$y=c_1{e}^{m_{1}x}+c_2{e}^{m_{2}x}$.

Case III: Conjugate Complex Roots

Ifm₁ and m₂ are complex roots, then $m_1=\alpha +i\beta$ and $m_2=\alpha -i\beta$, where $\alpha and\beta >0$ and are real and $i^2=-1$.

The general solution will be$$\begin{gathered} y=c_1{e}^{\alpha x}\cos \beta x+c_2{e}^{\alpha x}\sin \beta x \\ =e^{\alpha x}\left(c_1\cos \beta x+c_2\sin \beta x\right) \end{gathered}$$

Find the given differential equation

Consider the given differential equation $3y^{\text{'}\text{'}}+2y^{\text{'}}+y=0$.

Consider $y=e^{mx}$ as the solution of the differential equation $3y^{\text{'}\text{'}}+2y^{\text{'}}+y=0$.

Substitute $y=e^{mx},y\text{'}=m{e}^{mx}$ and $y\text{'}\text{'}=m^2{e}^{mx}$into $3y^{\text{'}\text{'}}+2y^{\text{'}}+y=0$to obtain the auxiliary equation.

$$\begin{gathered} 3m^2{e}^{mx}+2m{e}^{mx}+e^{mx}=0 \\ {e}^{mx}\left(3m^2+2m+1\right)=0 \end{gathered}$$

Solve the auxiliary equation as follows.

$3m^2+2m+1=0$

Determine the roots.

$m=\frac{-(-b)\pm \sqrt{b^2-4ac}}{2a}$

Therefore,

$$\begin{gathered} m_1=\frac{-\left(b\right)+\sqrt{b^2-4ac}}{2a} \\ =\frac{-\left(2\right)+\sqrt{-8}}{2\left(3\right)} \\ =\frac{-2+i·2\sqrt{2}}{2\left(3\right)} \\ =-\frac{1}{3}+i\frac{\sqrt{2}}{3} \end{gathered}$$

$$\begin{gathered} =\frac{-\left(2\right)+\sqrt{-8}}{2\left(3\right)} \\ =\frac{-2+i·2\sqrt{2}}{2\left(3\right)} \\ =-\frac{1}{3}+i\frac{\sqrt{2}}{3} \end{gathered}$$

And,

$$\begin{gathered} m_2=\frac{-\left(b\right)-\sqrt{b^2-4ac}}{2a} \\ =\frac{-\left(2\right)-\sqrt{-8}}{2\left(3\right)} \\ =\frac{-2-i·2\sqrt{2}}{2\left(3\right)} \\ =-\frac{1}{3}-i\frac{\sqrt{2}}{3} \end{gathered}$$

Compute the solution of the given differential equation

Therefore, solutions to the auxiliary equation are $\left(-\frac{1}{3}+i\frac{\sqrt{2}}{3},-\frac{1}{3}-i\frac{\sqrt{2}}{3}\right)$. The roots of the auxiliary equation are conjugate complex numbers.

For a differential equation which have complex roots, the roots are $m_1=\alpha +i\beta$and $m_2=\alpha -i\beta$.

Here, $m_1=-\frac{1}{3}+i\frac{\sqrt{2}}{3}$ and $m_2=-\frac{1}{3}-i\frac{\sqrt{2}}{3}$.

On comparison,$\alpha =-\frac{1}{3}$and$\beta =\frac{\sqrt{2}}{3}$.

Therefore, the general solution to the differential equation is,

$$\begin{gathered} y=e^{\alpha x}\left(c_1\cos \beta x+c_2\sin \beta x\right) \\ =e^{-\frac{1}{3}·x}\left(c_1\cos \left(\frac{\sqrt{2}}{3}\right)x+c_2\sin \left(\frac{\sqrt{2}}{3}\right)x\right) \\ =e^{-\frac{x}{3}}\left(c_1\cos \left(\frac{\sqrt{2}}{3}\right)x+c_2\sin \left(\frac{\sqrt{2}}{3}\right)x\right) \end{gathered}$$

Thus, the general solution is$\mathit{y}\mathbf{=}{\mathit{e}}^{\mathbf{-}\frac{\mathbf{x}}{\mathbf{3}}}\left(c_{1}cos\left(\frac{\sqrt{2}}{3}\right)x+c_{2}sin\left(\frac{\sqrt{2}}{3}\right)x\right)\mathit{y}\mathbf{=}{\mathit{e}}^{\mathbf{-}\frac{\mathbf{x}}{\mathbf{3}}}\left(c_{1}cos\left(\frac{\sqrt{2}}{3}\right)x+c_{2}sin\left(\frac{\sqrt{2}}{3}\right)x\right)$ .