(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find an equation that defines IMPLICITLY the parameterized family of solutions y(x) of the differential equation:

5xy dy/dx = x^{2}+ y^{2}

2. Relevant equations

y=ux

dy/dx = u+xdu/dx

C as a constant of integration

3. The attempt at a solution

I saw a similar D.E. solved using the y=ux substitution, but using it for mine wasn't as clean, so halfway through I decided to use an integrating factor:

5xy dy/dx = xDivide across by x (and switch the order of the LHS):^{2}+ y^{2}

dy/dx = [ x^{2}+ y^{2}] / 5xy

u + x du/dx = [ x^{2}+ u^{2}x^{2}] / 5x^{2}u

u + x du/dx = [ 1 + u^{2}] / 5u

du/dx + u/x = [ 1 + u^{2}] / 5ux

Integrating factor: (the integral symbol won't show up in the superscript)

Integrate both sides gets us:

e^{INT[1/x]}dx = e^{ln(x)}= x

du/dx ⋅ x + u/x ⋅ x = [ 1 + u^{2}] ⋅ x / 5ux

d/dx [ux] = [ 1 + u^{2}] / 5u dx

After this, it's just cleaning up so the u's are on one side. We get:

ux = x [ [ 1 + u^{2}] / 5u ] + C

We can now resubstitute the u's for y/x:

u - [ 1 + u^{2}] / 5u = C/x

[4u^{2}- 1] / 5u = C/x

[4[y/x]^{2}- 1] / 5[y/x] = C/x

4y/5x - x/5y = C/x

After this, I'm pretty stuck. I don't know how to isolate y(x) on one side. If I messed up anywhere, or if there is a much easier way to do this problem that I am ignoring, please let me know! Looking forward to the suggestions.

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# Finding an implicit solution to this differential equation

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