By the method of undetermined coefficients which of the foll

By the method of undetermined coefficients, which of the following Y is the correct form of a particular solution to the equation y⁽⁴⁾-3y⁽³⁾+2y\'\' = 4t-e^t+3e^et?

Correct answer: Y = At³+Bt²+Cte^t+De^3t

How do I reach this solution?

Solution

4t-e^t+3e^3t

Corresponding to 4t the guess usually is :At+B

Since t and 1 are already both solutions to homogeneous ode so we need to multipley this by t^2

So guess is

At^3+Bt^2

Corresponding to -e^t guess is: Ce^t but since e^t is solution to homogeneous ode so the guess is

Cte^t instead of Ce^t

Guess corresponding to 3e^3t is

De^{3t}

So correct form of particular solution is

Y=At^3+Bt^2+Cte^t+De^{3t}