# 2nd order inhomo differential eq.

## Homework Statement

$$y'' = t^2$$

## The Attempt at a Solution

The general solution to the homogenous diff equation is $$y(t)= C1 + tC2$$ i beliee. The particular solution is where i am having trouble.

the guess is of the form $$\alpha t^2 + \beta t + \gamma$$ ... but taking 2 deriatives leads to just $$2\alpha = t^2$$ ... this can only be zero if $$\alpha = \frac{t^2}{2}$$ but this is something i haven't encountered as the constants are usually numeric values... and what about the rest of constants of beta and gamma?

Mark44
Mentor
Try a particular solution of yp = At4.

The general solution will be the complementary solution (c1 + c2t) plus the particular solution.

Try a particular solution of yp = At4.

The general solution will be the complementary solution (c1 + c2t) plus the particular solution.

i am not following... i am trying to find a particular solution.... am i not supposed to find a polynomial of similar degree with unknown coefficients and finding derivates and then equating it to the regular diff equation to solve for the coefficients... trying a particular solution, i am unfamiliar with this method.

Mark44
Mentor
You need to look at nonhomogenous equations in two separate parts: the homogeneous equation and then the nonhomogeneous equation.

For your problem, the homogeneous problem is y'' = 0, to which the solutions are y = c1 + c2t.

Any linear combination of the functions 1 and t (i.e., any sum of constant multiples of 1 and t) will end up at 0 when you take the derivative twice, so a particular solution can't be 1, or t or any constant multiple of these.

One function whose 2nd derivative is t2 is t4, so for a particular solution to the nonhomogeneous problem, I suggest trying yp = At4. Plug this into the nonhomog. equation and solve for the value of A that works (this is called the method of undetermined coefficients, IIRC.

Your general solution to the nonhomog. equation will be y = c1 + c2t + At4, and your job is to figure out what A needs to be.

There's a lot more I could say, but it's probably in your textbook and you haven't come to it yet.

@projection don't believe when they say you are "guessing" an answer - it's not a guess it's a METHOD. If you build a template by the correct method it will ALWAYS give you a correct answer in constant-coefficient-linear DEs (that the method can handle) :)

You didn't build the correct template because you did it not by the METHOD.
I hope Mark44 explained it in a way you will understand :)