Find the explicit solution of the following initial value problem.

xy^2 dy/dx=y^3-x^3   ,   y(1)=2

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Start by dividing by y^3 both sides:

`(x/y)*(dy/dx) = 1 - (x/y)^3`

Put `x/y = t =gt x= ty =gt dx = (t'*y + t)dt =gt dt/dx = 1/(t'*y + t)`

`` `t/(t'*y + t) = 1 - t^3 =gt t'*y + t = t/(1 - t^3)`

`y*(dt/dy) = t/(1 - t^3) - t`  => `t*(dt/dy) = -t^4/(1 - t^3)`

Integrate both sides:

`int dy/y =- int (1 - t^3)/t^4`

`` `ln |y| = - t^-3/-3 + ln |t|`

`ln |y| = -1/(3t^3)- ln |t| + c`

`` `ln |y|+ ln |t| = -1/3t^3 + c`

`` `ln |yt| = -1/3t^3 + c`

`` `ln |x| = -1/3(x/y)^3 + c`  =>`x = 1/e^(3(x/y)^3)`

ANSWER: `x = 1/e^(3(x/y)^3)`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial