Solve the initialvalue problem d2ydt2 10 dydt 25 y 0 y1

Solve the initial-value problem d^2y/dt^2 + 10 dy/dt + 25 y = 0, y(1) = 0, y\'(1) = 1.

Solution

This is a linear homogeneous recurrence

assume solution of the form: y=exp(kt) Subsituting gives

k^2+10k+25=0

k=-5 and repeated roots

So,

y=exp(-5t)(A+Bt)

y(1)=exp(-5t)(A+B)=0

So, A=-B

y=B exp(-5t)(-1+t)

y\'(t)=B exp(-5t)-5Bexp(-5t)(-1+t)

y\'(1)=B exp(-5)=1

B=exp(5)

y= exp(5-5t)(-1+t)