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Differential equation of form y'=f(y/x)

Posted March 5th, 2008 by Structure

Differential equation of the form

$$y'(x)=f(\frac{y(x)}{x})$$

where f is a continuous function can easily be solved by reducing it to an equation with separable variables.
Take $ y(x)=xu(x) $ where u is a new function. Then we have

$$y'(x)=xu'(x)+u(x)$$

Now we can write

$$xu'(x)+u(x)=f(u(x))$$

or

$$xu'(x)=f(u(x))-u(x)$$

We can write

$$\frac{u'(x)}{f(u(x))-u(x)}=\frac{1}{x}$$

and also

$$\int_{x_0}^x\frac{u'(t)}{f(u(t))-u(t)}dt=\int_{x_0}^x\frac{1}{t}dt$$
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