MHB Randy's question at Yahoo Answers (linear differential quation)

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The discussion revolves around solving the linear differential equation (1 + a^2)q' + 6aq = a/(1 + a^2)^3. The homogeneous solution is derived as y = C(1 + x^2)^{-3} using separation of variables. The variation of parameters method is then applied, leading to the expression for C(x) as C(x) = (1/2)log(1 + x^2) + c. Ultimately, the general solution is presented as y = c/(1 + x^2)^3 + log(1 + x^2)/(1 + x^2)^3. This solution provides a comprehensive answer to Randy's query on Yahoo Answers.
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Randy's question at Yahoo! Answers (linear differential equation)

Here is the question:

What is the general Solution of
(1 + a^2)q' + 6aq = a/(1 + a2)^3,
Where q is a function of a? you can switch q with y and a with x if that is easier. Please show work
thank you

Here is a link to the question:

What is the general solution of the differential Equation? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
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It is a linear equation. Let's solve the homogeneous (1+x^2)\dfrac{dy}{dx}+6xy=0:
\dfrac{dy}{y}+\dfrac{6x}{1+x^2}\;dx=0\\<br /> \log |y|+3\log (1+x^2)=K\\<br /> \log |y|=K-3\log (1+x^2)\\<br /> y=C(1+x^2)^{-3}

Now, we use the variation of parameters method. Substituing y=C(x)(1+x^2)^{-3} in the original equation:

(1+x^2)[C&#039;(x)(1+x^2)^{-3}+C(x)(-3)(1+x^2)^{-3}2x]+6xC(x)(1+x^2)^{-3}=x(1+x^2)^{-3}

Symplifying: C&#039;(x)=\dfrac{x}{1+x^2} hence C(x)=\dfrac{1}{2}\log (1+x^2)+c. As a consequence the general solution of the given equation is

y=\dfrac{c}{(1+x^2)^3}+\dfrac{\log (1+x^2)}{(1+x^2)^3}
 
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