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#### Pranav

##### Well-known member

- Nov 4, 2013

- 428

**Problem:**

Solve:

$$\frac{x\,dx-y\,dy}{x\,dy-y\,dx}=\sqrt{\frac{1+x^2-y^2}{x^2-y^2}}$$

**Attempt:**

I rewrite the given differential equation as:

$$\frac{(1/2)d(x^2-y^2)}{x^2d(y/x)}=\sqrt{\frac{1+x^2-y^2}{x^2-y^2}}$$

I thought of using the substitution $x^2-y^2=t^2$ but that doesn't seem to help. The following is what I get after the substitution:

$$\frac{t\sqrt{x^2-t^2}\,dt}{t\,dx-x\,dt}=\sqrt{1+t^2}$$

But I don't see how to proceed from here.

Any help is appreciated. Thanks!