User:Egm4313.s12.team8.jmurillo

Problem Statement
'''Kreyszig 2011 Pg. 59 Problem 4''' Find a general solution for the following ODE.

Given: $$\displaystyle y''+4y'+(\pi^2+4)y=0 $$

Solution
The characteristic equation for the given ODE is  $$\displaystyle \lambda^2+\lambda+(\pi^2+4)=0 $$

The roots of the characteristic equation are complex roots in the form $$\displaystyle \lambda=-\frac{1}{2}a\pm i \omega $$

The roots of the characteristic equation are

$$\displaystyle \lambda=\frac{1}{2}(-4\pm 2 \pi i) $$

Therefore this ODE falls under Case III which has a general solution in the form

$$\displaystyle y=e^\frac{-ax}{2} (Acos \omega x+Bsin \omega x) $$

The general solution to the ODE is

$$\displaystyle y= e^{-2x} (Acos\pi x+Bsin\pi x) $$