How do i deal with this separable differential equation?

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June 19th 2010, 01:45 PM
Jones

How do i deal with this separable differential equation?

Hello,

I need to solve this separable differential equation

$4y^3\frac{dy}{dx}=x^{-2}~~~~~,y(1) = 1$

We have $4y^3~dy = \frac{1}{x^2}dx$

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

But what is $\bigg(-\frac{1}{x}\bigg)^{1/4}$ ?
June 19th 2010, 01:49 PM
mr fantastic

Quote:

Originally Posted by Jones

Hello,

I need to solve this separable differential equation

$4y^3\frac{dy}{dx}=x^{-2}~~~~~,y(1) = 1$

We have $4y^3~dy = \frac{1}{x^2}dx$

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

But what is $\bigg(-\frac{1}{x}\bigg)^{1/4}$ ?

The correct answer is $y^4 = - \frac{1}{x} + C$ . Then you substitute y(1) = 1 to get the value of C. Then you make y the subject (if that's what the question expects).
June 20th 2010, 12:52 AM
Jones

This is confused,

Don't you want the answer in terms of y?
June 20th 2010, 02:48 AM
e^(i*pi)

You want y to be the subject usually. ie: y = f(x).

Find C using the way Mr F said: $1 = -\frac{1}{1}+C$

Once the value of C is known you can take the 4th root (obviously substitute in the value of C calculated in the previous step) : $y = \sqrt[4]{-\frac{1}{x}+C}$

As you said you cannot take the 4th root of a negative number so you will have domain restrictions in your answer since $-\frac{1}{x} +C \geq 0$

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