Higher order ODE

Higher order ODE are little tricky to solve and can be classified into homogeneous and non homogeneous.

a normal OD will be of the format below -

Y'''+const.Y''+const.Y'+Y = G(X)

when G(X) = 0 equation is called homogeneous otherwise when G(X) not equal 0 then it is non homogeneous.

Basic method to solve both the equations is same however change will be introduced depending upon type of equation.

Let us first look at homogeneous solution-

we have equation -

Y''+const.Y'+Y = 0 .......................1

assume Y = exp(rt)

then Y' = r*exp(rt)

Y'' = r*r*exp(rt)

using this equation (1) reduces to

exp(rt) (r*r + const.*r+1) = 0 .................................2

exp(rt) can't be zero

r*r+const.*r+1 = 0 can be solved for roots of the equation

Solution of ODE will look like -

Y = c1*exp(r1*t) + c2*exp(r2*t) where r1 and r2 are roots of equation (2)