Differential equation problem

**gracey** · March 22nd 2009, 02:13 AM

I need to find the equation of the curve that satisfies the differential equation dy/dx = y / x(x-1) and passes through the point (1,2)

To start i worked out that

1/y dy/dx = 1 / x(x-1)

i tried to integrate and got

lny = ln x(x-1) but i am not sure if that is right or what else to do

thanks

**running-gag** · March 22nd 2009, 02:22 AM

Hi

Your integration on x side is not correct

$\int \frac{1}{x(x-1)}dx = \int \left(\frac{1}{x-1} - \frac{1}{x}\right)dx = \ln \frac{|x-1|}{|x|}$

**The Second Solution** · March 22nd 2009, 02:25 AM

Originally Posted by gracey

I need to find the equation of the curve that satisfies the differential equation dy/dx = y / x(x-1) and passes through the point (1,2)

To start i worked out that

1/y dy/dx = 1 / x(x-1)

i tried to integrate and got

lny = ln x(x-1) but i am not sure if that is right or what else to do

thanks

Did you differentiate $\ln x(x - 1)$ as a check? Do you get back $\frac{1}{x(x - 1)} ?$ (If you answer yes to this question then you need to seriously review differentiation).

The right hand side must be integrated using a partial fraction decomposition: $\frac{1}{x (x - 1)} = \frac{1}{x - 1} - \frac{1}{x}$ .

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