Using Matrix and Abel's formula to find second solution

[Trillionaire]

Homework Statement

There is a Differential Equations question that I am stuck:

Consider the equation: $(t^2)(y'')-2(t)(y')+(t^2+2)y=0$, where t>0

It is given that y1= t*cos(t) is a solution

(a) There are two formulations for determining the Wronskian, (1) based on the determinant of a matrix which involved a pair of fundamental solutions and their derivatives, and (2) in terms of Abel's formula. Allowing the constant in Abel's formula to be equal to one, equate the two expressions for the Wronskian to derive a first order differential equation for the second solution y2.

(b) Solve the first order equation for this second solution, y2. Note: $d/dt (tan(t)) = 1/(cos^2(t))$

Homework Equations

Abel's theorem: Wronskian = $c*exp(-\int p(t)\,dt)$for y'' + p(t)y' + q(t)y =0
Matrix method: Wronskian = (y1)(y2') - (y1')(y2)

The Attempt at a Solution

For the Wronskian I got W = (t*cos(t))(y2') - (cos(t) - t*sin(t))(y2). If I did my derivatives right, that should be correct. For Abel's formula, I got W = e^(t^2). However, I'm not sure if it's possible to solve equation (t*cos(t))(y2') - (cos(t) - t*sin(t))(y2) = e^(t^2). I think I did something wrong, but not sure what. Can someone help?

 

[Adrmay]

Also, for part (b), I'm not sure how to approach this. I'm not sure how to use the tan(t) derivative in the equation. Can someone help? Thanks.