Given y 2y 6t 5e2t find the correct from to use yp if the

Given y\'\' - 2y = 6t + 5e^2t find the correct from to use y_p if the equation is solved using under coffin cute do not solve the equation.

Solution

For getting correct form for the yp based on undetermined coefficients the correct form is guessed based on the inhomogeneous part. In this problem it is

6t+5e^{2t}

We can seperately guess form for each component ie for 6t and then for 5e^{2t}

For polynomial 6t the form is: At+B

For exponential form is : C e^{2t}

Since ode is linear we can just add them to get the correct form which is

At+B+Ce^{2t}