Does This Differential Equation Solution Satisfy Given Conditions?

[goaliejoe35]

Homework Statement
Verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

y=C1+C2 lnx

xy'' + y' = 0

y=0 when x=2
y'=(1/2) when x=2

The attempt at a solution

Here's what I did so far...

y=C1+C2 lnx
y'=C1+C2(1/x)
y''=C1+C2(-1/x^2)

Since xy''+y'=0 I then substituted y' and y'' into the equation.

x(C1+C2(-1/x^2))+(C1+C2(1/x))=0

After this step I am stuck. If you could help push me in the right direction that would be great. Also could you verify that what I already did is right?

 

[kreil]

The derivative of a constant is zero, as I'm sure you're aware, so C1 drops out of y' and y''

then since $y'+xy''=0$ you have $$\frac{C2}{x}-\frac{C2x}{x^2}=\frac{C2}{x}-\frac{C2}{x}=0$$ as desired.

The second part is just plugging in some numbers, $(2)(y'')+\frac{1}{2}=0 \implies y''= - \frac{1}{4}$