Second order differential equations of the homogen type

y” (x)+ a y'(x) + by(x) = 0

are possible to solve with the aid of the characteristic equation

r² + a r +b =0

If this have the roots r₁ and r₂  

the solution is given by

y(x) = Ce^r1x + De^{r2 x}  

If the equation is inhomogenous and the right side is a polynom assume a solution a polynomial of the same degree

If the right side is a trigonometric function assume a as a solution a combination of trigonometric functions.

Ex. Solve the equation y” -3y – 4y = 0

Solve the characteristic equation: r^{2-3r-4 = 0
r=4 och r = -1.}

The general solution is y = Ce^-4x + De^x
Where C and D are arbetare constants.