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markosheehan September 5th, 2017 12:56 PM

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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.

skipjack September 5th, 2017 02:22 PM

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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