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Supposedly simple calculus problem.

  • #1
210
1

Homework Statement



Solve the differential equation.

[itex]\frac{dy}{dx}[/itex] = [itex]\frac{1}{xy}[/itex] + [itex]\frac{y}{x}[/itex]


The Attempt at a Solution



[itex]\frac{dy}{dx}[/itex] = [itex]\frac{x + xy^{2}}{x^{2}y}[/itex]

[itex]\frac{dy}{dx}[/itex] = ([itex]\frac{x}{x^{2}}[/itex])([itex]\frac{1 + y^{2}}{y}[/itex])

([itex]\frac{y}{1 + y^{2}}[/itex])dy = [itex]\frac{x}{x^{2}}[/itex]dx

[itex]\oint\frac{y}{1 + y^{2}}[/itex]dy = [itex]\oint\frac{x}{x^{2}}[/itex]dx

I can't seem to get any further than there
 
Last edited:

Answers and Replies

  • #2
lurflurf
Homework Helper
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change variables u=y/x
 
  • #3
33,262
4,962

Homework Statement



Solve the differential equation.

[itex]\frac{dy}{dx}[/itex] = [itex]\frac{1}{xy}[/itex] + [itex]\frac{y}{x}[/itex]


The Attempt at a Solution



[itex]\frac{dy}{dx}[/itex] = [itex]\frac{x + xy^{2}}{x^{2}y}[/itex]
It's simpler to just factor out 1/x, giving
dy/dx = (1/x)(1/y + y) = (1/x)(1 + y2)/y
[itex]\frac{dy}{dx}[/itex] = ([itex]\frac{x}{x^{2}}[/itex])([itex]\frac{1 + y^{2}}{y}[/itex])

([itex]\frac{y}{1 + y^{2}}[/itex])dy = [itex]\frac{x}{x^{2}}[/itex]dx

[itex]\int\frac{y}{1 + y^{2}}[/itex]dy = [itex]\int\frac{x}{x^{2}}[/itex]dx

I can't seem to get any further than there
The first integral can be done with an ordinary substitution. In the integral on the right, simplify x/x2.
 
  • #4
Ray Vickson
Science Advisor
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Try multiplying both sides of your original DE by y.

RGV
 
  • #5
33,262
4,962
change variables u=y/x
Try multiplying both sides of your original DE by y.

RGV
Guys, the equation is separable, and the OP is close to getting a solution. IMO, the best thing to do is help him along the path he's on (which is the simplest), rather than steering him in a different direction.
 
  • #6
Ray Vickson
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Guys, the equation is separable, and the OP is close to getting a solution. IMO, the best thing to do is help him along the path he's on (which is the simplest), rather than steering him in a different direction.
I could not see the rest of his submission on my i-phone at the coffee shop, so that's why I replied.

RGV
 

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