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- #1

- Feb 5, 2012

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Continuing from >>this<< thread, my friend gave me another question. He wants to check his work specifically on parts a) and d). So I will only focus on those parts.

**Question Summary:**

Given the Riccati equation,

\[\frac{dy}{dx}+e^{-x}y^2+y+e^x=0\]

we have to solve it using the substitution,

\[y(x)=\frac{e^x}{w}\frac{dw}{dx}\]

a) Show that,

\[\frac{dy}{dx}=e^x\left[\frac{1}{w}\frac{d^{2}w}{dx^2}-\frac{1}{w^2}\left(\frac{dw}{dx}\right)^2+\frac{1}{w}\frac{dx}{dx}\right]\]

b) Show the differential equation for \(w(x)\) is,

\[\frac{d^{2}w}{dx^2}+2\frac{dw}{dx}+w=0\]

c) Find the general solution of the differential equation you found in part b).

d) Use \(\displaystyle y(x)=\frac{e^x}{w}\frac{dw}{dx}\) to find the general solution of \(y(x)\) of the original Riccati equation. Simplify this expression as much as possible. You should be able to combine the two integration constants from part c) into one single integration constant.