Find the solution to the following differential equations. y″ − 6y′ + 25y = 2 sin(t/2) – cos(t/2) ?
- stanschimLv 71 month agoFavourite answer
The auxiliary equation is r^2 - 6r + 25 = 0. This yields roots of 3+4i and 3-4i.
The homogeneous solution is therefore y = e^3t(Acos(4t) + Bsin(4t)).
To this we must add the particular solution to get the general solution.
Using the method of undetermined coefficients, we assume a solution of the form
y = Asin(t/2) + Bcos(t/2)
y' = (1/2)Acos(t/2) - (1/2)Bsin(t/2)
y'' = -(1/4)Asin(t/2) - (1/4)Bcos(t/2)
Plugging these into the original equation gives:
(-1/4)Asin(t/2) - (1/4)Bcos(t/2) - 3Acos(t/2) + 3Bsin(t/2) +25Asin(t/2) + 25Bcos(t/2) = 2sin(t/2) - cos(t/2)
Combining like terms and equating terms on each side gives:
[-(1/4)A + 3B + 25A](sin(t/2)) + [-3A - (1/4)B + 25B](cos(t/2)) = 2sin(t/2) - cos(t/2)
[(99/4)A + 3B](sin(t/2)) + [-3A + (99/4)B](cos(t/2)) = 2sin(t/2) - cos(t/2)
A = 56/663 and B = -20/663 meaning the particular solution is:
y = (56/663)sin(t/2) - (20/663)cos(t/2).
The general solution is:
y = e^(3t)(Acos(4t) + Bsin(4t)) + (56/663)sin(t/2) - (20/663)cos(t/2), where A and B are constants to be determined by initial conditions.
- Wayne DeguManLv 71 month ago
With explanations, the solution to this involves two sides of A4.
I have a solution written out, but fear that it will receive little thanks or acknowledgement.
Give me an encouraging reason to post.
- 1 month ago
Answer to this question is