# Verify a solution

• February 9th 2010, 10:40 PM
ChingKing08
Verify a solution
Verify that $xy=lny +c$ is a solution of the differential equation $dy/dx=(y^2)/(1-xy)$ for every value of the constant c.

I can't figure this one out...I don't even know where to start. Some help would be much appreciated. Thanks!
• February 9th 2010, 10:49 PM
Black
Use implicit differentiation on $xy=\text{ln}y+c$.
• February 9th 2010, 10:51 PM
dedust
Quote:

Originally Posted by ChingKing08
Verify that $xy=lny +c$ is a solution of the differential equation $dy/dx=(y^2)/(1-xy)$ for every value of the constant c.

I can't figure this one out...I don't even know where to start. Some help would be much appreciated. Thanks!

we need to show that $xy=lny +c$ satisfy DE $dy/dx=(y^2)/(1-xy)$

start from $xy=lny +c$, differentiate it

$y + xy' = \frac{y'}{y}$

$y(y + xy') = y'$

$y^2 + xyy' = y'$

$y^2 = y'(1 - xy)$

$y' = \frac{y^2}{1-xy}$

so the equation $xy=\ln y +c$ satisfy the differential equation $dy/dx=(y^2)/(1-xy)$