For each of the following forcing functions gt write down th

For each of the following forcing functions g(t), write down the test function with the fewest terms which can be used to obtain a particular solution via the Method of Undetermined Coefficients. Do not solve for the constants. g(t) = t^2. g(t) = cos(t) + 2t sin(t) g(t) = t^2 e^t/2 cos(t/2) + e^t/2

Solution

(i)

g(t)=t^2

If the forcing function is a polynomial of degree n then the guess for particular solution is a general polynomial of degree n with unknown coefficients

So the test function here is:

At^2+Bt+C

ii)

For cos(t) or sin(t) the guess is: A sin(t)+B cos(t)

If t cos(t) or t sin(t) is present guess is: (At+B) sin(t)+(Ct+D) cos(t)

So the test function is:

(At+B) sin(t)+(Ct+D) cos(t)

iii)

The test function is:

(At^2+Bt+C)e^{t/2}sin(t)+(Dt^2+Et+F) e^{t/2}cos(t)+Ge^{t/2}