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