just because i needed to refresh myself on this particular method .... variation of parameters.

Question

amistre64 · Answer

Variation of Parameters:

the solution to the differential equation:

$$y'' + py' + qy = f(x)$$
can be determined if the homogenous solution $y_h$ is known.
$$(y_h)'' + p(y_h)' + q(y_h) = 0$$

Let $y_h=c_1u(x)+c_2v(x)$ such that $c_1$ and $c_2$ are arbitray constants, given $u(x)$ and $v(x)$ are linearly independant.

By varing the parameters of $y_h$, we can determine a function $y_p$.

Let g(x) and h(x) be 2 arbitraty functions and replace the arbitrary constants with them to obtain:
$$y_p=g(x)u(x)+h(x)v(x)~:=~gu+hv$$

Take the first and second derivatives to insert them into the equation:

$$\begin{matrix}(y_p)' &=& g'u + gu' + h'v+ hv'\
&=& gu' + hv' + (\underbrace{g'u + h'v}_{=0})
\end{matrix}$$

$$(y_p)''= g'u' +gu'' + h'v' + hv'' + (\underbrace{g'u + h'v}_{=0})'$$

By setting $g'u + h'v = 0$ we give ourselves the ability to chose g or h at random; and if one is chosen the other can be determined (substitution method of solving a system of equations).  Also, setting it equal to zero frees up some calculations, otherwise we would have ended up with 8 terms in the 2nd derivative.

Now we can input these into the original equation:

$$\begin{pmatrix}
+q~gu & +q~hv &+g'u' \
+p~gu' & +p~hv' &+h'v' \
+gu'' & +p~hv'' & \
\color{red}{=0}&\color{red}{=0}

\end{pmatrix}=f(x)$$

In the first 2 columns, we can factor out g and h, and what remains is a product that contains the homogeneous solutions -- the solutions that equates to 0 -- so that our setup reduces to simply:

$$g'u'+h'v' = f(x)$$
Recall that we defined something similar to this beforehand:
$$g'u + h'v = 0$$

We now have 2 equations in 2 unknowns (g' and h') that can be worked out using algebraic methods: substitution, cramers rule, elimination, etc.
$$g'u + h'v = 0\
 g'u' + h'v' = f(x)$$

When we know g' and h', we integrate to obtain g and h which were the arbitrary functions that we were looking for.

thank you herbert gross;  will post his video once i find it on the google again.

amistre64 · Answer

http://ocw.mit.edu/resources/res-18-008-calculus-revisited-complex-variables-differential-equations-and-linear-algebra-fall-2011/part-ii/lecture-5-variations-of-parameters/