Question

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

$y\text{'}\text{'}\text{'}+5y\text{'}\text{'}+3y\text{'}-9y=0$

Short answer

The general solution for the differential equation with x as the independent variable is $y\left(x\right)=c_1{e}^x+c_2{e}^{-3x}+c_{3}x{e}^{-3x}$

Step-by-step solution

Auxiliary equation:

Consider the equation .$y\text{'}\text{'}\text{'}+5y\text{'}\text{'}+3y\text{'}-9y=0$

The associated auxiliary equation is $r^3+5r^2-3r-9=(r-1){(r+3)}^2=0$which has $r=-3$as a double solution and$r=1$ as a simple solution.

General solution:

The general solution to the given equation is given by

$y\left(x\right)=c_1{e}^x+c_2{e}^{-3x}+c_{3}x{e}^{-3x}$

Hence the final solution is $y\left(x\right)=c_1{e}^x+c_2{e}^{-3x}+c_{3}x{e}^{-3x}$