Solving a Nasty First Order ODE

[nichevo]

Homework Statement

This problem is from Blanchard "Differential Equations" Chapter one review, question 32.

$${\frac {d}{dt}}y \left( t \right) -{\frac {y \left( t \right) {t}^{3}}<br /> {1+{t}^{4}}}=2$$

The Attempt at a Solution

Using an integrating factor yields:

$${\frac {d}{dt}} \sqrt [4]{1+{t}^{4}}y \left( t \right) =2\,\sqrt [4]{1+{t}^{4}}$$

This is unworkable...

Any hints would be greatly appreciated. I suspect that I am overlooking a simple guess.

 

[Ben Niehoff]

Yeah, you're making it much harder than it is. Just multiply out by $(1+t^4)dt$, and you get

$$(1+t^4)dy - yt^3dt = 2(1+t^4)dt$$

Hint: try to make the left side look like the Quotient Rule. What is the derivative of $(1+t^4)$?