Question

Solve the given differential equation.$\mathbf{25}{\mathit{x}}^{\mathbf{2}}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{25}\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

The solution for the differential equation is $y=c_1\text{cos}\left(\frac{1}{5}\text{ln}x\right)+c_2\text{sin}\left(\frac{1}{5}\text{ln}x\right)$

Step-by-step solution

Define Differential equation

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

Solve differential equation

Consider,

$y\text{'}=\frac{dy}{dx}$

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

Substitute in the above equation,

$25x^2\frac{{\text{d}}^{2}y}{\text{d}{x}^2}+25x\frac{\text{d}y}{\text{d}x}+y=0$

Let,

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

Then the derivatives become,

$\frac{\text{d}y}{\text{d}x}=m{x}^{m-1}$

$\frac{{\text{d}}^{2}y}{\text{d}{x}^2}=m\left(m-1\right)x^{m-2}$

So, the equation is:

$$\begin{gathered} 25x^2\frac{{\text{d}}^{2}y}{\text{d}{x}^2}+25x\frac{\text{d}y}{\text{d}x}+y=0 \\ 25x^{2}m\left(m-1\right)x^{m-2}+25xm{x}^{m-1}+x^m=0 \end{gathered}$$

Keep $x^m$ commonly out,

$\begin{array}{rcl}{x}^m\left(25m\left(m-1\right)+25m+1\right)& =& 0\\ x^m\left(25m^2-25m+25m+1\right)& =& 0\\ x^m\left(25m^2+1\right)& =& 0\\ x^m\left(5m+i\right)\left(5m-i\right)& =& 0\end{array}$

Solving for m, by equal to 0:

$\left(5m+i\right)\left(5m-i\right)=0$

$m_1=-\frac{i}{5}$ ; $m_2=\frac{i}{5}$

Use Case 3: Conjugate complex roots

Then the general solution becomes,

$y=x^{\alpha }\left[c_1\text{cos}\left(\beta \text{ln}x\right)+c_2\text{sin}\left(\beta \text{ln}x\right)\right]$

Where the real parts of complex solution is $\alpha =0$ and positive complex$\beta =2$ as these are the conjugate complex roots of the equation.

After substituting the values of$\alpha \mathrm{and}\beta$ , the solution is $y=c_1\text{cos}\left(\frac{1}{5}\text{ln}x\right)+c_2\text{sin}\left(\frac{1}{5}\text{ln}x\right)$ .