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June 17th, 2017, 12:32 PM  #1 
Member Joined: Jun 2017 From: India Posts: 63 Thanks: 3  A beginner differential equation question
What is a differential equation ? It is an equation involving an unknown function (solution) and its derivatives . 
June 17th, 2017, 12:51 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 17,919 Thanks: 1383 
The function (of x) is y, and dy/dx is its derivative (with respect to x).

June 17th, 2017, 04:33 PM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,647 Thanks: 680 
If you are asking whether $\displaystyle y= \pm\sqrt{\frac{2}{3}x^3+ C}$ really does satisfy $\displaystyle \frac{dy}{dx}= \frac{x^2}{y}$, it is easy to check. First, squaring both sides gives $\displaystyle y^2= \frac{2}{3}x^3+ C$ and differentiating both sides with respect to x, $\displaystyle 2y\frac{dy}{dx}= 2x^2$. Dividing both sides by 2y, $\displaystyle \frac{dy}{dx}= \frac{x^2}{y}$, precisely the differential equation you started with. Or, more directly but more complicated, differentiating the original formula with respect to x, $\displaystyle \frac{dy}{dx}= \pm\frac{1}{2}\left(\frac{2}{3}x^3+ C\right)^{1/2}(2x^2)= \frac{x^2}{\pm\sqrt{\frac{2}{3}x^3+ C}}= \frac{x^2}{y}$. Last edited by skipjack; June 17th, 2017 at 05:16 PM. 
June 17th, 2017, 10:17 PM  #4 
Member Joined: Jun 2017 From: India Posts: 63 Thanks: 3 
Thanks , Found some good materials here National Council Of Educational Research And Training :: Home 

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