Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Find two-parameter families of solutions for the following differential equation $$ yy'' + (y')^2 = 1. $$

Can this be explained on the appropriote steps required to find the solution of $y = \pm \sqrt{x^2+Fx+G}$, where $F$ and $G$ are constants.

share|cite|improve this question
up vote 2 down vote accepted

The key observation is the fact that $$\dfrac{d}{dx}\left( y \dfrac{dy}{dx}\right) = y \dfrac{d^2 y}{dx^2} + \left( \dfrac{dy}{dx}\right)^2$$ Hence, we have that $$\dfrac{d}{dx}\left( y \dfrac{dy}{dx}\right) = 1$$ i.e. $$y \dfrac{dy}{dx} = x + c_1 \implies y^2 = x^2 + 2c_1x + c_2 \implies y = \pm \sqrt{x^2 + Fx + G}$$

share|cite|improve this answer
    
For this OP I would probably add $$ \dfrac{d}{dx}\left( y^2 \right) = 2 y \dfrac{dy}{dx} = 2 x + 2 c_1 $$ – Will Jagy Nov 18 '12 at 20:53

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.