2nd order inhomo differential eq.

[projection]

Homework Statement

[tex]y'' = t^2[/tex]

The Attempt at a Solution

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

the guess is of the form [tex]\alpha t^2 + \beta t + \gamma[/tex] ... but taking 2 deriatives leads to just [tex]2\alpha = t^2[/tex] ... this can only be zero if [tex]\alpha = \frac{t^2}{2}[/tex] 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]

Try a particular solution of y_p = At⁴.

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

 

[projection]

Try a particular solution of y_p = 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]

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 y_p = 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.

 

[gomunkul51]

@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 :)