Q8E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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

 $2y\text{'}\text{'}\text{'}+5y\text{'}\text{'}-13y\text{'}+7y=0$

Step-by-Step Solution

Answer

The general solution for the differential equation with x as the independent variable is  $y\left(x\right)=c_1{e}^x+c_{2}x{e}^x+c_3{e}^{-7x}$

Step 1: Auxiliary equation:

Considering the equation

 $y\text{'}\text{'}\text{'}+5y\text{'}\text{'}-13y\text{'}+7y=0$

 

The associated auxiliary equation is  $r^3+5r^2-13r\text{'}+7={(r-1)}^2(r+7)=0$ which has a double solution and r = -7 as a simple solution.

Step 2: General solution:

The general solution to the equation is  $y\left(x\right)=c_1{e}^x+c_{2}x{e}^x+c_3{e}^{-7x}$

Hence the final solution is  $y\left(x\right)=c_1{e}^x+c_{2}x{e}^x+c_3{e}^{-7x}$