Suppose we have a second order linear equation of the form y

Suppose\' we have a second order linear equation of the form y\" + p(t)y\'+ q(t)y = 0. and we know that y_1(t) = t + 2 and yi(t) = e^-2t are solutions. Find the solution satisfying the initial conditions y(0) = 2. y\'(0) = -2.

Solution

will The trivial solution willbe y(t)=(2+t) e^-2t as thee solutions are

given to0 be t+2 and e^-2t which satisfies the boundary conditions thatisy(0)=(2+0)e⁰=2 and y^/ (t) =(2+t)x-2e^-2t+(0+1)e^-=--2e ⁰+1.e⁰=-1

is

e