Solving ODE: 2y(1+x^2√y)dx + x(2+x^2√y)dy = 0 | Step-by-Step Guide

[AdrianZ]

Homework Statement

2y(1+x^2\sqrt{y})dx + x(2+x^2\sqrt{y})dy = 0

The Attempt at a Solution

well, I substituted x^2√y=u but then when I tried to differentiate it I understood it would be so hard. Please check and see if I've differentiated it correctly:

√y = u/x^2 -> y = u^2.x^-4 -> dy/dx = 2u.u'x^(-4) - 4x^(-5).u^2
Is that correct? if yes, then I think I've just made the problem harder. how can I solve that ODE?

 

[ehild]

It is correct. Substitute for y and y' in the original equation, simplify and arrange: it is a separable ODE.

ehild

 

[phyzguy]

You could try grouping the terms into perfect differentials. For example, 2 y dx + 2 x dy = 2 d(xy). The other two terms can be written similarly as d(something) . Then write u = xy and v = something and it is a simple linear ODE.