The idea of variation of parameters is to seek the solution to

$\displaystyle \bf{x}' = A\bf{x} + \bf{b} $

in the form

$\displaystyle

\bf{v}_1(t)\bf{x}_1(t) + \bf{v}_2 (t)\bf{x}_2 (t) = F(t)\bf{v}(t) $

whereby $\displaystyle F(t) = \begin{bmatrix} x_1 & x_2 \end{bmatrix} $ is a fundamental matrix.

Note that the constant vector

**c** of coefficients is replaced by an unknown function

**v**(t).

What is the equation for our unknown function

**v**(t) you ask?

It is fairly straight forward to derive:

$\displaystyle

(F\bf{v})' = F'\bf{v} + F\bf{v'} = A(F\bf{v}) + \bf{b} \implies

F\bf{v'} = \bf{b} \implies

\bf{v'} = F^{-1}\bf{b} $ .

*IMPORTANT:* You must make sure that $\displaystyle F^{-1} $ exists in order to solve for

**v** !!!! (not factorial(Happy))

Finally, using integration, we have:

$\displaystyle

\bf{x}(t) = F(t) \int F(t)^{-1} \bf{b}(t)dt

$

I'll now rewrite the original problem in latex so it is more palatable to the eye:

Find the general solution of

$\displaystyle

\bf{x}' = A\bf{x} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} $ , whereby $\displaystyle A = \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix} $.

Does this help?