Q12E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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

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

Step-by-Step Solution

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 1: 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.

 

Step 2: 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  $y\left(x\right)=c_1{e}^x+c_2{e}^{-3x}+c_{3}x{e}^{-3x}$