User:Egm6321.f09.team3.mooney/HW4

= Problem 6: Modified King 1.1.a b =

Given
$$ (x-1)y'' - xy' + y = x $$ $$xy'' + 2y' + xy = x $$

Find
The general solution for a and b using 1) Variation of Parameter (assuming you know $$u_1$$ and $$u_2$$ 2) Alternative Method from the notes

Solution
First, solving part a by method 1) Variation of Parameters

It is known that $$u_1 = e^x $$

And, $$u_2 = -x$$

Now we have the full solution in the form: $$ y(x) = c_1(x)*e^x - c_2(x)*x $$

Since this is the full solution 2 assumptions have to be made, $$c_1'*u_1 + c_2'*u_2 = 0$$

And therefore, $$c_1'*u_1' + c_2'*u_2' = \frac{x}{x-1}$$

Now we check the Wronskian:

$$\bar W = u_1 u_2 ][u_1' u_2' = e^x -x][e^x -1$$

$$W = det(\bar W) = \frac{1}{e^{x}*(x-1)}$$

Since the Wronskian is not 0, the solutions $$c_1'$$ and $$c_2'$$ exist.

where

$$ c_1'][c_2' = \bar W ^{-1} 0][f = -1 x][-e^x e^x ^{-1} 0][\frac{x}{(x-1)} $$

So the solution looks like

$$c_1' = \frac{x^2}{e^{x}*(x-1)}$$

and $$c_2' = \frac{x}{x-1}$$

Integrate to get $$c_1$$ and $$c_2$$

$$ c_1 = ...something I missed I'm guessing ...$$

$$ c_2 = ln{x-1} + x $$