Find the general solution to y4 6y3 9y 0SolutionWe have a

Find the general solution to y(4) 6y(3) + 9y = 0.

We have a linear homogeneous recurrence relation with constant coefficients

So solution is of the form

y=exp(kx)

Substituting gives

k^4-6k^3+9k^2=0

k^2(k-3)^2=0

So, two roots, k=0 and k=3 both repeated roots

If ,k=r is a reapeated root teh general solutoin corresponding to it is

e^{rx}(A+Bx)

So corresponding to k=0 we have: e^{0}(A+Bx)=A+Bx

Corresponding to k=3 we ahve: e^{3x}(C+Dx)

Hence general solution is

y(x)=A+Bx+e^{3x}(C+Dx)