Q5E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Find a general solution for the differential equation with x as the independent variable:

$y^m+3y^n+28y\text{'}+26y=0$

Step-by-Step Solution

Answer

The general solution for the differential equation with x as the independent variable is   $y=c_1{e}^{-x}+c_2{e}^{-x}\cos 5x+c_3{e}^{-x}\sin 5x$

Step 1: Auxiliary equation:

The auxiliary equation for  $y\text{'}\text{'}\text{'}+3y\text{'}\text{'}+28y\text{'}+26=0$ is   $r^3+3r^2+28r+26=0$. The solution of previous equation is:

 

 $\begin{array}{c}{r}^3+3r^2+28r+26=0\\ (r+1)(r^2+2r+26)=0\\ r=-1\\ r^2+2r+26=0\\ r_{2,3}=\frac{-2\pm \sqrt{2^2-4.26}}{2}\\ =-1\pm 5i\end{array}$

 

Where I is imaginary unit

Step 3: General solution:

Then  $\alpha =-1,\beta =5$ and we can write the general solution for the differential equation: 

 $y=c_1{e}^{-x}+c_2{e}^{-x}\cos 5x+c_3{e}^{-x}\sin 5x$

Hence the final solution is  $y=c_1{e}^{-x}+c_2{e}^{-x}\cos 5x+c_3{e}^{-x}\sin 5x$