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September 5th, 2017, 12:56 PM   #1
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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.
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Last edited by skipjack; September 5th, 2017 at 01:43 PM.
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September 5th, 2017, 02:22 PM   #2
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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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