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**1.Homework Statement**

We know the derivation of the method of variation of parameters for second order

**scalar**differential. The task is to derive the method of variation of parameters for scalar equations using this approach: first convert the scalar equation into the

**first order system**and then apply the method of variation of parameters for

**systems**.

*Hint: Assume that the fundamental set of solutions {y*

_{1}(t), y_{2}(t)} for your scalar equation is known. How would you construct a fundamental matrix for your system from the scalar fundamental set?## Homework Equations

(my work, not in problem statement)

y''+p(t)y'+q(t)y=g(t) <--Second order nonhomogenous scalar

y' = P(t)y + g(t) <--first order nonhomogenous system

## The Attempt at a Solution

y''+p(t)y'+q(t)y=g(t)

change of variables...

z(t)= [ z

_{1}(t)

z

_{2}(t) ]

z

_{1}(t) = y(t)

z

_{2}(t) = y'(t)

z'

_{2}(t)+p(t)z

_{2}(t)+q(t)z

_{1}(t)=g(t)

z'

_{1}(t) = y'(t) = z

_{2}(t)

z'

_{2}(t) = y''(t) = -z

_{2}(t)p(t)-z

_{1}(t)q(t)+g(t)

P(t) =

[ 0 1

-q(t) -p(t) ]

G(t) =

[ 0

g(t) ]

y' = P(t)y + g(t) <--now what I have has this form (first order system)

Now I have the variation of parameters for systems derivation in front of me, the end result of which is:

y(t) = [tex]\psi[/tex](t)u

_{0}+ [tex]\psi[/tex](t)[tex]\int_t_0^t[/tex][tex]\psi[/tex]

^{-1}(s)g(s)ds

So I guess the hard part now is finding what the fundamental matrix [tex]\psi[/tex] is. I know the fundamental set is {y

_{1}, y

_{2}}

In the variation of parameters for second order differential equations we make the assumption y

_{1}u'

_{1}+ y

_{2}u'

_{2}= 0. I am not supposed to make this assumption of this problem.