First order differential equation question

[idks16]

The problem is : dy/dx=(x(x^2+1))/4y^3 when y(0)=-1/√2
This is my work so far:
∫4y^3dy=∫x(x^2+1)dx
(y^4)/2=((x^2+1)^2)/2+c
The answer from the textbook is y=-(√(x^2+2)/2)
As you can see, my work will never equal the textbook answer when you put it in the y= stuff form. What did I do wrong?

 

[grep6]

I got a slightly different answer than what you posted from the text

y(x) = -\sqrt{\frac{1}{2}(x^2+1)}

and mathematica agrees with me, so perhaps a typo?

Anyway, it looks like your on the right track, although go back through the integration, I think you may be off by a factor.
Then apply the boundary condition to find the integration constant.
And simplify the algebra down to the answer.
Also be conscience of taking roots,
y(x) = ±(stuff)^{1/4}
good luck