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September 5th, 2017, 11:56 AM  #1 
Member Joined: May 2016 From: Ireland Posts: 96 Thanks: 1  equation
Solve (d^2)y/d(x^2) =y given that dy/dx =1 and y=1 when x=0. I am not sure how to do this. What should the limits for v dv be? Question 1 is the worked answer to the question. I am not sure where they get v=1, as it is not given in the question. Last edited by skipjack; September 5th, 2017 at 12:43 PM. 
September 5th, 2017, 01:22 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,846 Thanks: 1567 
The image is difficult to read. When x = 0, v = dy/dx = 1. As $v = \dfrac{dy}{dx}$, $y = \dfrac{d^2y}{dx^2} = \dfrac{dv}{dx} = \dfrac{dv}{dy}\cdot\dfrac{dy}{dx} = v\dfrac{dv}{dy}$. Integrating $y = v\dfrac{dv}{dy}$ with respect to $y$ gives ${\small\dfrac12}v^2 = {\small\dfrac12}y^2 + c$, where $c$ is a constant. Note: it's asserted that $v = y$ won't work, but that's incorrect. Is that sufficient help? 

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