# Need help with this initial-value problem

• Jan 31st 2010, 10:42 AM
steph3824
Need help with this initial-value problem
Find the solution of the given initial value problem in explicit form.

y' = x(x²+1) / 4y³, y(0) = -1 /√2

According to my textbook, the correct answer is y= -√((x²+1)/2)

Here is what I am doing and hopefully someone can tell me where I'm going wrong. I can see that it is separable, so I separated it and integrating both sides I got y^4=1/4x^4 + 1/2x² + C. Solving for C I got 1/4, however I am not sure that any of this is correct so far. Anyways, I know I'll never get the right answer unless I get the correct implicit solution and the correct value of C, so help me out with those if what I have written is incorrect!
• Jan 31st 2010, 10:49 AM
TheEmptySet
Quote:

Originally Posted by steph3824
Find the solution of the given initial value problem in explicit form.

y' = x(x²+1) / 4y³, y(0) = -1 /√2

According to my textbook, the correct answer is y= -√((x²+1)/2)

Here is what I am doing and hopefully someone can tell me where I'm going wrong. I can see that it is separable, so I separated it and integrating both sides I got y^4=1/4x^4 + 1/2x² + C. Solving for C I got 1/4, however I am not sure that any of this is correct so far. Anyways, I know I'll never get the right answer unless I get the correct implicit solution and the correct value of C, so help me out with those if what I have written is incorrect!

Okay so you have (It is correct) Just keep going

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

Now lets simplify the right hand side a bit

$\displaystyle y^4=\frac{1}{4}(x^4+2x^2+1)=\frac{1}{4}(x^2+1)^2$

Now take the 4th roots and simplify.(don't forget the plus or minus) The intial conditions which root you need to take to satisfy the equaiton.

I hope this helps(Wink)
• Jan 31st 2010, 11:24 AM
steph3824
That was definitely helpful...I was able to get to the correct answer after you showed the best way to simplify it, thanks!!