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echomochi
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Homework Statement
Find an equation that defines IMPLICITLY the parameterized family of solutions y(x) of the differential equation:
5xy dy/dx = x2 + y2
Homework Equations
y=ux
dy/dx = u+xdu/dx
C as a constant of integration
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 = x2 + y2
dy/dx = [ x2 + y2 ] / 5xy
u + x du/dx = [ x2 + u2x2 ] / 5x2u
u + x du/dx = [ 1 + u2 ] / 5u
Divide across by x (and switch the order of the LHS):dy/dx = [ x2 + y2 ] / 5xy
u + x du/dx = [ x2 + u2x2 ] / 5x2u
u + x du/dx = [ 1 + u2 ] / 5u
du/dx + u/x = [ 1 + u2 ] / 5ux
Integrating factor: (the integral symbol won't show up in the superscript)
eINT[1/x]dx = eln(x) = x
du/dx ⋅ x + u/x ⋅ x = [ 1 + u2 ] ⋅ x / 5ux
d/dx [ux] = [ 1 + u2 ] / 5u dx
ux = x [ [ 1 + u2 ] / 5u ] + C
u - [ 1 + u2 ] / 5u = C/x
[4u2 - 1] / 5u = C/x
[4[y/x]2 - 1] / 5[y/x] = C/x
4y/5x - x/5y = 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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