Question

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

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

Short answer

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

Step-by-step solution

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.

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