Differential Equations, Separable, Simplification of answer

1. Jun 29, 2015

RyanTAsher

1. The problem statement, all variables and given/known data
I believe I have solved this differential equation, yet do not know how the book arrived at it's answer...

Solve the differential equation in its explicit solution form.

The answer the book gives is...

2. Relevant equations

Separable Differential Equation

3. The attempt at a solution

dy/dx = x(x^2+1)/(4y^3)

(4y^3)dy = (x^3+x)dx

∫(4y^3)dy = ∫(x^3+x)dx

y^4 = 1/4x^4 + 1/2x^2 + c

(initial condition, y(0) = -1/sqrt(2))

(-1/sqrt(2))^(4) = 0 + 0 + c

C = -1/4
....

y^4 = 1/4x^4 + 1/2x^2 - 1/4

y = (1/4x^4 + 1/2x^2 - 1/4)^(1/4)

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I've experimented with simplifying this a bit and found a few other ways to express it, but nothing like what the book has written as the answer.

2. Jun 29, 2015

RUber

Your sign on C is wrong.

$(\frac{x^4}{4}+\frac{x^2}{2} + \frac 14 ) = (\frac{x^2}{2} + \frac 12 )^2$

3. Jun 29, 2015

SammyS

Staff Emeritus
First. You made a mistake in finding C. What is (-1/sqrt(2))4 ? Fixing that will allow some factoring in the resulting expression.

4. Jun 29, 2015

RyanTAsher

Oh wow, I don't think I would have seen that factor regardless. Thank you though. That helped a lot.

5. Jun 29, 2015

RyanTAsher

Thank you, I understand now. In regards to the -, out front the answer from the book, I understand that comes from the square root, but how do they determine whether to go with the - or + solution. I haven't learned intervals of validity within the book yet...

6. Jun 29, 2015

RUber

The radical implies the positive. Your initial condition forces the negative choice.

7. Jul 4, 2015

Dr. Courtney

Before rejecting an answer, you should plug it into the diff eq and see if it works.