write down the form of the particular solution to y''+p(t)y'+q(t)y=g(t),where g(t)=16(e^(7t))sin(10t),g(t)=(9(t^2)-10t)cos(t),g(t)=-(e^(-2t))(3-5t)sin(10t).

Question

just finding particular solution by using 3 option of g(t) equations.

Answer 1

For the particular solution, we need to take the product of the particular solutions of the exponential, polynomial and trigonometric functions separately. Constants are shown in uppercase letters.

 

1. g(t)=(16e^7t)sin(10t): y=(e^7t)(Asin(10t)+Bcos(10t)). The constant 16 is absorbed into the constants A and B.

2. g(t)=(9t^2-10t)cos(t): y=(At^2+Bt+C)sin(t)+(Dt^2+Et+F)cos(t).

3. g(t)=-e^-2t(3-5t)sin(10t) (this was submitted separately in a question on 25 Dec 2016):

y=(e^-2t)(A+Bt)sin(10t)+(e^-2t)(C+Dt)cos(10)t.

If the solution to the homogeneous DE y"+p(t)y'+q(t)y=0 yields similar terms, the particular solution needs to be multiplied by t or even t^2 to avoid nonsensical solutions.

Of course, I could be completely wrong! Happy New Year!