Supposedly simple calculus problem.

[Darth Frodo]

Homework Statement

Solve the differential equation.

\frac{dy}{dx} = \frac{1}{xy} + \frac{y}{x}

The Attempt at a Solution

\frac{dy}{dx} = \frac{x + xy^{2}}{x^{2}y}

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

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

\oint\frac{y}{1 + y^{2}}dy = \oint\frac{x}{x^{2}}dx

I can't seem to get any further than there

 

[lurflurf]

change variables u=y/x

 

[Mark44]

Homework Statement

Solve the differential equation.

\frac{dy}{dx} = \frac{1}{xy} + \frac{y}{x}

The Attempt at a Solution

\frac{dy}{dx} = \frac{x + xy^{2}}{x^{2}y}

It's simpler to just factor out 1/x, giving
dy/dx = (1/x)(1/y + y) = (1/x)(1 + y²)/y

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

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

\int\frac{y}{1 + y^{2}}dy = \int\frac{x}{x^{2}}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.

 

[Ray Vickson]

Try multiplying both sides of your original DE by y.

RGV

 

[Mark44]

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.

 

[Ray Vickson]

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