Question

Solve the given differential equation.${\mathit{x}}^{\mathbf{2}}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{7}\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{41}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

The required solution is $y=x^4\left[c_1\text{cos}\left(5\text{ln}x\right)+c_2\text{sin}\left(5\text{ln}x\right)\right]$.

Step-by-step solution

Define conjugate complex roots

If the conjugate complex root$m_1=\alpha +i\beta ,m_2=\alpha -i\beta$ , are real, here$\alpha and\beta >0$ , a solution can be obtained by using the formula,

$y=C_1{x}^{\alpha +i\beta }+C_2{x}^{\alpha -i\beta }$

Hence, the conjugate of Euler’s formula can be,

$$\begin{gathered} x^{i\beta }=cos\left(\beta lnx\right)+isin\left(\beta lnx\right) \\ {x}^{-i\beta }=cos\left(\beta lnx\right)-isin\left(\beta lnx\right) \end{gathered}$$

By deriving the conjugate complex roots and Euler’s formula, the general solution can be obtained as,

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

Solve the given differential equation by using the general solution formula

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

Substitute the values of$y,y\text{'},y\text{'}\text{'}$in the equation,

$x^{2}y\text{'}\text{'}-7xy\text{'}+41y=x^2\left[m(m-1)x^{m-2}\right]-7x\left[m{x}^{m-1}\right]+41x^m=0$

Simplify the equation,

$m{x}^m(m-1)-7m{x}^m+41x^m=0$

$m^2{x}^m-m{x}^m-7m{x}^m+41x^m=0$

$m^2{x}^m-8m{x}^m+41x^m=0$

$x^m\left(m^2-8m+41\right)=0$

Use the quadratic formula, to solve the characteristics equation $\left(m^2-8m+41\right)=0$,

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Replace $x=m_{1,2}$, $a=1,b=-8,c=41$, as:

$\begin{array}{rcl}{m}_{1,2}& =& \frac{-\left(-8\right)\pm \sqrt{{\left(-8\right)}^2-4\left(1\right)\left(41\right)}}{2\left(1\right)}\\ m_{1,2}& =& \frac{8\pm \sqrt{-100}}{2}\\ & =& \frac{8\pm 10i}{2}\\ & =& 4\pm 5i\\ m_1& =& 4+5i;m_2=4-5i\end{array}$

Thus, use the general solution of a complex conjugate roots,

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

Since, the conjugate complex roots are$m_1=4+5i;m_2=4-5i$ , the values of$\alpha =4;\beta =5$

Substitute the values in the general solution,

$y=x^4\left[c_1\text{cos}\left(5\text{ln}x\right)+c_2\text{sin}\left(5\text{ln}x\right)\right]$

Therefore, the solution for the given differential equation is$y=x^4\left[c_1\text{cos}\left(5\text{ln}x\right)+c_2\text{sin}\left(5\text{ln}x\right)\right]$ .