Consider the following Riccatti equation:(adsbygoogle = window.adsbygoogle || []).push({});

dy/dx= -y^2+ a(x)y + b(x) (Eq. 2)

Here a(x) and b(x) are arbitrary functions.

1. Set y(x)= u'(x)/ u(x) where u(x) is a function to be determined. Use (Eq. 2) to show that u(x) satisfies a linear differential equation of second order.

2. Set a=0 and b=1. Solve (Eq. 2) by separating variables using y(0)=0 as the initial condition.

3. Find the function u(x) corresponding to the case a=0, b=1 and y(0)=0. When solving the second order equation for u(x) assume that u'(0)= y(0) and u(0)= 1. Compare the solution of (Eq. 2) from Part 2 with u'(x)/u(x).

4. Eplain why it is nice to replace a non-linear first order differential equation with a linear second order one.

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for part 1 here is what i have substituting y=u'/u

(dy/dx)= -(u'/u)^2+a(x)(u'/u)+b(x)

multiply everything by u to get:

u(dy/dx)=-u(u')^2+a(x)(u')+b(x)u

So am i done here or is there more??? (I have a really lousy professor who

just barely got his phD and cant really teach. therefore i am having a lot of diffuculty understanding the matrial. i feel like i am teaching myself ODE....so please help)

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# Homework Help: Ricatti equation

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