Solving a second order ODE using reduction of order

[Bonnie]

Homework Statement

Hi there, I have an assignment which involves using reduction of order to solve for a second solution to an ode (the one attached). However this is a method I am new to, and though I have tried several times, I'm somehow getting something wrong because the LHS and RHS are not matching up, that is, when I substitue in the solution I have found, the RHS does not equal zero as it should.

Homework Equations

The Attempt at a Solution

I have attached my working (Sides 1 and 2), if anyone could point out what I'm doing wrong it would be greatly appreciated, this is driving me nuts!

 

[Delta2]

According to wolfram your general solution is correct.

You do a blunt mistake in the check, you put $y'=-\frac 1 2 t^{-\frac 1 2}$, the minus in front is not needed. It is clear that $y'=\frac{1}{2} t^{-\frac 1 2}$ for $y=t^{\frac 1 2}$

 

[Bonnie]

According to wolfram your general solution is correct.

You do a blunt mistake in the check, you put $y'=-\frac 1 2 t^{-\frac 1 2}$, the minus in front is not needed. It is clear that $y'=\frac{1}{2} t^{-\frac 1 2}$ for $y=t^{\frac 1 2}$

Oh, that was dumb. Thank you!