Solve DE Using Substitution and Integration Techniques | Helpful Tips Included

[TogoPogo]

The problem states:

"By using the substitution $y=xu$, show that the differential equation $\frac{dy}{dx}=\frac{y+\sqrt{x^{2}+y^{2}}}{x}, x>0$ can be reduced to the d.e. $x\frac{du}{dx}=\sqrt{u^{2}+1}$.

Hence, show that if the curve passes through the point $(1,0)$, the particular solution is given by $y=\frac{1}{2}(x^{2}-1)$."

I managed to get the d.e. into the form $x\frac{du}{dx}=\sqrt{u^{2}+1}$ but I have no idea how to integrate $\frac{du}{\sqrt{u^{2}+1}}$. Wolfram Alpha is giving me some inverse hyperbolic sine stuff which I haven't learned yet (I'm in high school). All I've really 'learned' from my teacher so far was solving separable DE's, and inseparable DE's with $y=ux$, however some of the questions that we were given required other techniques like integrating factors and stuff. Is this DE a special case or something?

Anyways, how would I approach this? Do I square both sides to get rid of the square root sign?

Many thanks.

 

[vela]

Try the substitution u=tan v.

 

[TogoPogo]

I got it! Thank you. I didn't know how to integrate secx but Wolfram helped out.