# Supposedly simple calculus problem.

• Darth Frodo
In summary, the conversation focuses on solving a differential equation in the form of dy/dx = 1/(xy) + y/x. The attempt at a solution involves factoring and substitution, but ultimately the equation is separable and can be solved by multiplying both sides by y. The participants in the conversation suggest helping the original poster continue on their current path rather than suggesting a different approach.

## 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

Last edited:
change variables u=y/x

Darth Frodo said:

## 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
Darth Frodo said:
$\frac{dy}{dx}$ = ($\frac{x}{x^{2}}$)($\frac{1 + y^{2}}{y}$)

($\frac{y}{1 + y^{2}}$)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.

Try multiplying both sides of your original DE by y.

RGV

lurflurf said:
change variables u=y/x

Ray Vickson said:
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.

Mark44 said:
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