Second-Order Linear Differential Equation

Differential equation of the form is called second-order linear differential equation. When , then it is homogeneous, otherewise it is nonhomogeneous. First we will solve homogeneous equations with constant coefficients, that is with and where and are two constant. The so called characteristic equation of the second-order linear homogeneous differential equation

is . We first find roots of the characteristic equation using Maple command solve . Depending on nature of the roots will have three different cases:

REAL and DIFFERENT roots: If and are two different real roots then the soltion is given by .

REAL and EQUAL roots: If , then the solution is given by

.

COMPLEX roots: If is a complex root, where and are its real and imaginary part respectively, then the solution is given by . We first define RD (RD stands for real and differenent), RE (RE stands for real and equal) and C (C stands for complex) operators to faciliate our calculations.

> restart;

> RD:=c[1]*exp(r[1]*x)+c[2]*exp(r[2]*x);RE:=c[1]*exp(r[1]*x)+c[2]*r[1]*exp(r[1]*x);C:=exp(u*x)*(c[1]*cos(v*x)+c[2]*sin(v*x));

Example 1: Solve .

Example 2: Solve .

Example 3: Solve .

Example 4: Solve the differential equation from Example 3, that satisfies the initial conditions .