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What kind of differential equation do you use 'variation of parameters' to solve?

I'm just struggling to understand what the method of 'variation of parameters' and when it is necessary to use it. Could anyone explain this to me? Is it used if and only if the equation is in the form d2ydx2+p(x)dydx+q(x)y=f(x)\frac{d^{2}y}{dx^{2}}+p(x)\frac{dy}{dx}+q(x)y=f(x), where p,q,fp,q,f are functions, or is it only possible to use it for equations in the form ax2d2ydx2+bxdydx+cy=f(x)ax^{2}\frac{d^{2}y}{dx^{2}}+bx \frac{dy}{dx}+cy=f(x), where a,b, and c are constants?
(edited 9 years ago)
Original post by Brian Moser
I'm just struggling to understand what the method of 'variation of parameters' and when it is necessary to use it. Could anyone explain this to me? Is it used if and only if the equation is in the form d2ydx2+p(x)dydx+q(x)y=f(x)\frac{d^{2}y}{dx^{2}}+p(x)\frac{dy}{dx}+q(x)y=f(x), where p,q,fp,q,f are functions, or is it only possible to use it for equations in the form ax2d2ydx2+bxdydx+cy=f(x)ax^{2}\frac{d^{2}y}{dx^{2}}+bx \frac{dy}{dx}+cy=f(x), where a,b, and c are constants?

It works for all ODEs. As long as you have a basis of the space of homogeneous solutions (that is, complementary functions), variation of parameters will work to find a particular solution.

EDIT: as Hasufel points out, though, there are often much less painful ways to do a given differential equation.
(edited 9 years ago)
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Hasufel
16
your last equation is called a Cauchy_euler equation - solved with the substitution x=etx=e^{t}

Variation of parameters is (usually) used when the forcing term (the function on the right-hand side of the diff eqn) is not of a form which can be used in the method of undetermined coefficients (sin, cos, tan, e^(f(x)), a polynomial..) These all work because the finding of a solution using the appropiate "trial" method works for these.

For instance, something like the other trig functions, sec, csc, cot (etc)

Matbe this`ll help..
(edited 9 years ago)
vary of params.pdf110.9KB

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