Homogeneous linear equations with constant coefficients

Second -order differential equations
1-Homogeneous linear equations with constant coefficients
A second order homogeneous equation with constant coefficients is written as
ay^ +by^ +cy=0 where a, b and c are constant.
Let us summarize the steps to follow in order to find the general solution:
1. Write down the characteristic equation am^2+bm+c=0
2. Find the roots of the characteristic equation m_(1 ) and m_2
Here we have three cases
I. If m_(1 ) and m_2 are distinct real numbers (m_(1 )=m_2 ) (this happens if b^2-4ac>0 ),
then the general solution is
y_h=c_1 e^(m_1 x)+c_2 e^(m_2 x)
II. If m_(1 )=m_2 (this happens if b^2-4ac=0 ), then the general solution is
y_h=(c_1 x+c_2 ) e^mx
III. If m_(1 ) and m_2 are conjugate complex numbers (m_(1 )=¯(m_2 )=?+?i)
(this happens if b^2-4ac<0 ), then the general solution is
y_h=e^?x (c_1 sin??x+c_2 cos??x )
Example 1: Find the general solution of y^ -6y +8y=0
Solution
Characteristic equation and its roots
m^2-6m+8=0 ? (m-2)(m-4)=0 ? m_1=2 and m_2=4
The general solution is
y_h=c_1 e^2x+c_2 e^4x

Example 2: Find the general solution of y^ +y^ -20y=0
Solution
m^2+m-20=0 (Characteristic equation)
(m+4)(m-5)=0 ? m_1=-4 and m_2=5 (The roots)
y_h=c_1 e^(-4x)+c_2 e^5x (General solution)