SOLUTION: (s") + 6s' - 9s = t^2 where s" is like d^2x/dt^2 (s") + 6s' - 9s = t^2 where s" is like d^2x/dt^2

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Question 199856: (s") + 6s' - 9s = t^2
where s" is like
d^2x/dt^2
(s") + 6s' - 9s = t^2
where s" is like
d^2x/dt^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
This is a 2nd order linear differential equation. So here are the steps to finding the general solution:


Step 1) Find the complementary solution s%5Bc%5D

Solve the characteristic equation r%5E2%2B6r-9=0 to get r=-3%2B3%2Asqrt%282%29 or r=-3-3%2Asqrt%282%29. Since we have two unique real roots, this means that the complementary solution is


Note: recall, if you have two real roots r%5B1%5D and r%5B2%5D of the characteristic equation, then the complementary solution is s%5Bc%5D=c%5B1%5D%2Ae%5E%28r%5B1%5D%2Ax%29%2Bc%5B2%5D%2Ae%5E%28r%5B2%5D%2Ax%29


Step 2) Find the operator that annihilates the right hand side t%5E2


The operator D%5E3 annihilates 1, t, and t%5E2


We can rewrite the given differential equation as: D%5E2%2B6D-9=t%5E2


When we apply the operator D%5E3 to both sides, we get: D%5E3%28D%5E2%2B6D-9%29=0


Because D%5E3 annihilates 1, t, and t%5E2, we know that the particular solution s%5Bp%5D is


s%5Bp%5D=A%2BBt%2BCt%5E2


Derive s%5Bp%5D to get:






Derive again to get:




Now plug this information into the original differential equation to get


2C%2B6%28B%2B2Ct%29-9%28A%2BBt%2BCt%5E2%29=t%5E2


2C%2B6B%2B12Ct-9A-9Bt-9Ct%5E2=t%5E2


-9Ct%5E2%2B%2812Ct-9Bt%29%2B%28-9A%2B6B%2B2C%29=t%5E2


So this means that

-9C=1

12C-9B=0

-9A%2B6B%2B2C=0


Solve the system above to get A=-10%2F81, B=-4%2F27, and C=-1%2F9


So the particular solution is s%5Bp%5D=-10%2F81-%284%2F27%29x-%281%2F9%29x%5E2



This means that the general solution is


s=s%5Bc%5D%2Bs%5Bp%5D


Plug in s%5Bc%5D and s%5Bp%5D (what we found earlier) to get:





So that is the final answer.



Note: there is another way to solve this problem, but it involves nasty integrals. Even though there's a lot going on here, the solution method is really straightforward.