Solving Higher Order ODEs: y''''''+y'''=t

[mshiddensecret]

Homework Statement

y''''''+y'''=t

Homework Equations

The Attempt at a Solution

I got all the roots and solved the homo eq.

Then I tried to guess the partial eq and got At+B

However, I don't know how to proceed because the 6th derivative or the 3rd would be 0.

 

[SteamKing]

Homework Statement

y''''''+y'''=t

Homework Equations

The Attempt at a Solution

I got all the roots and solved the homo eq.

Then I tried to guess the partial eq and got At+B

I think you mean, you tried to guess the particular solution and got At + B

However, I don't know how to proceed because the 6th derivative or the 3rd would be 0.

It's not clear why you guessed y_p = At + B, since the highest order derivative is 6. This implies that y_p should be a 7th degree polynomial.

 

[LCKurtz]

You don't need a 7th degree polynomial for $y_p$ for this problem. Try $y_p = Ct^4$.

 

[Ray Vickson]

Mod note: removed a quote that was too much help.

You can also let $z(t) = y'''(t)$ and write the DE as $(z(t) - t)''' + (z(t)-t) = 0$, which is homogeneous of degree 3 in $z(t)-t$. After finding $z(t)$, integrating three times (with constants of integration included) will get $y(t)$.

 

[haruspex]

I think you mean, you tried to guess the particular solution and got At + B
It's not clear why you guessed y_p = At + B, since the highest order derivative is 6. This implies that y_p should be a 7th degree polynomial.

It only implies the general solution will be of degree 5, no? The degree of the particular solution will often be the sum of the least degree of differentiation and the highest degree of the polynomial on the other side of the equation. In this case, 3+1=4.