# Verify a solution

• Feb 9th 2010, 10:40 PM
ChingKing08
Verify a solution
Verify that $\displaystyle xy=lny +c$ is a solution of the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$ 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!
• Feb 9th 2010, 10:49 PM
Black
Use implicit differentiation on $\displaystyle xy=\text{ln}y+c$.
• Feb 9th 2010, 10:51 PM
dedust
Quote:

Originally Posted by ChingKing08
Verify that $\displaystyle xy=lny +c$ is a solution of the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$ 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 $\displaystyle xy=lny +c$ satisfy DE $\displaystyle dy/dx=(y^2)/(1-xy)$

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

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

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

$\displaystyle y^2 + xyy' = y'$

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

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

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