# Thread: General solution to differential equation

1. ## General solution to differential equation

Hi! I'm having trouble with this problem on my homework. Any help you can give would be wonderful!! Thanks!

Consider the differential equation:
.
a) Find the general solution to the above differential equation. (Instruction: Write the answer in a form such that its numerator is 1 and its integration constant is --- rename your constant if necessary.)

b) Find the particular solution of the above differential equation that satisfies the condition at .

2. Originally Posted by jsl
Hi! I'm having trouble with this problem on my homework. Any help you can give would be wonderful!! Thanks!

Consider the differential equation:
.
a) Find the general solution to the above differential equation. (Instruction: Write the answer in a form such that its numerator is 1 and its integration constant is --- rename your constant if necessary.)

b) Find the particular solution of the above differential equation that satisfies the condition at .
I am just going to rewrite this as

$y'=y^2\left(x^3-x\right)$ if you don't mind.

Then $\frac{y'}{y^2}=x^3-x$ So

$\int\frac{y'}{y^2}dx=\int\left[x^3-x\right]dx$. Evaluating gives

$\frac{-1}{y}=\frac{x^4}{4}-\frac{x^2}{2}+C$

Solving gives $y=\frac{-4}{x^4+2x^2+C_1}$

3. actually:

(a) $4/(t^4-2t^2+4C)$
(b) $4/(t^4-2t^2+1)$