y t 4 y t 5 y t 5 t e2t First find a solution to the hom

y^\" [t] - 4 y^\' [t] + 5 y [t] = 5 t + e^2t First find a solution to the homogeneous equation, then use variation of parameters to find a particular solution to the given equation.

Solution

homogeneous equation

y\'\'(t) - 4y\'(t) + 5y(t) = 0

assume solution e^{mt}

where m is the solution of equation

x^2 - 4x + 5

(x-2)^2 = -1

m = 2 + i or m = 2-i

so 2 independent general solutions are

y = e^2t cost and y = e^2t sint