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

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

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) (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

Solve the characteristic equation to get or . Since we have two unique real roots, this means that the complementary solution is

Note: recall, if you have two real roots and of the characteristic equation, then the complementary solution is

Step 2) Find the operator that annihilates the right hand side

The operator annihilates , , and

We can rewrite the given differential equation as:

When we apply the operator to both sides, we get:

Because annihilates , , and , we know that the particular solution is

Derive to get:

Derive again to get:

Now plug this information into the original differential equation to get

So this means that

Solve the system above to get , , and

So the particular solution is

This means that the general solution is

Plug in and (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.

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