Question

Solve the given differential equation.${\mathit{x}}^{\mathbf{2}}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{3}\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

The solution for the differential equation is$y=c_1{x}^{2+\sqrt{6}}+c_2{x}^{2-\sqrt{6}}$

Step-by-step solution

Define Differential equation

Anequation with one or more derivatives of a function. The derivative of the function is given by $\frac{\text{d}y}{\text{d}x}$is known as Differential equation.

Solve differential equation

Consider,

$y\text{'}=\frac{\text{d}y}{\text{d}x}$

$y\text{'}\text{'}=\frac{{\text{d}}^{2}y}{\text{d}{x}^2}$

Substitute in the above equation,

$x^2\frac{{\text{d}}^{2}y}{\text{d}{x}^2}-3x\frac{\text{d}y}{\text{d}x}-2y=0$

Let,

$y=x^m$ ; $y\text{'}=m{x}^{m-1}$; $y\text{'}\text{'}=m(m-1)x^{m-2}$

Then the derivatives become,

$$\begin{gathered} \frac{\text{d}y}{\text{d}x}=m{x}^{m-1} \\ \frac{{\text{d}}^{2}y}{\text{d}{x}^2}=m(m-1)x^{m-2} \end{gathered}$$

So, the equation is:

$$\begin{gathered} x^2\frac{{\text{d}}^{2}y}{\text{d}{x}^2}-3x\frac{\text{d}y}{\text{d}x}-2y=0 \\ {x}^{2}m(m-1)x^{m-2}-3xm{x}^{m-1}-2x^m=0 \end{gathered}$$

Keep $x^m$commonly out,

$$\begin{gathered} x^m\left(m\right(m-1)-3m-2)=0 \\ {x}^m\left(m^2-m-3m-2\right)=0 \\ {x}^m\left(m^2-4m-2\right)=0 \end{gathered}$$

Elaborating the equation,

$$\begin{gathered} x^m\left(m^2-2·2·m+2^2-2^2-2\right)=0 \\ {x}^m\left({(m-2)}^2-6\right)=0 \\ {x}^m(m-2-\sqrt{6})(m-2+\sqrt{6})=0 \end{gathered}$$

Solving for m,

$\left(m-2-\sqrt{6}\right)\left(m-2+\sqrt{6}\right)=0$

Using Factorize method,

$m_1=2+\sqrt{6}$ and$m_2=2-\sqrt{6}$

Use Case 1: Distinct real roots

Then the general solution becomes,

$y=c_1{x}^{m_1}+c_2{x}^{m_2}$

After substituting the value of$m_1\mathrm{and}{m}_2$ , the solution is$y=c_1{x}^{2+\sqrt{6}}+c_2{x}^{2-\sqrt{6}}$ .