The equation is y5y6y10e2x part a Find the general solutions

The equation is y\'\'+5y\'-6y=10e^2x (part a)

Find the general solutions of the equation y\" + 5y\' - 6y = 10e2x by the annihilator approach. Solve the system of differential equations by an annihilator approach d2x/dt2 - dy/dt = t dx/dt + dy/dt = -3x - 3y + 2

Solution

x(t) = -1/2 c_3 e^(-3 t) (2 e^t+1) (e^t-1)^2+c_2 e^(-t) (e^t-1)+1/2 c_1 e^(-3 t) (3 e^(2 t)-1)+t^2/2-t+5/3
y(t) = -3/2 c_1 e^(-3 t) (e^(2 t)-1)-c_2 e^(-t) (e^t-1)+1/2 c_3 e^(-3 t) (-3 e^(2 t)+2 e^(3 t)+)+3)-t^2/2+t-1