2nd order differential equation using reduction of order

[efghi]

Homework Statement

Use the method of reduction of order to find a second solution of the given differential equation

t²y'' - 4ty' + 6y, t>0; y₁(t) = t²

The Attempt at a Solution

Here's what I have so far:
y = vt²
y' = 2tv + t²v'
y'' = 2v + 4tv' + t²v''

so

t² (2v + 4tv' + t²v'') - 4t (2tv + t²v') + 6(vt²) = 0

t⁴v'' = 0

I'm stuck here. Am I supposed to divide by y, which would make it t²v'' = 0
wouldn't that make v'' = 0 anyways? What do I do now? I know I'm supposed to integrate it some how but I am just utterly lost from here...

PLEASE HELP!

 

[Hootenanny]

Homework Statement

Use the method of reduction of order to find a second solution of the given differential equation

t²y'' - 4ty' + 6y, t>0; y₁(t) = t²

The Attempt at a Solution

Here's what I have so far:
y = vt²
y' = 2tv + t²v'
y'' = 2v + 4tv' + t²v''

so

t² (2v + 4tv' + t²v'') - 4t (2tv + t²v') + 6(vt²) = 0

t⁴v'' = 0

I'm stuck here. Am I supposed to divide by y, which would make it t²v'' = 0
wouldn't that make v'' = 0 anyways? What do I do now? I know I'm supposed to integrate it some how but I am just utterly lost from here...

You're absolutely spot on here. Let's look at your final line

t⁴v''=0

Now the boundary conditions state that t is positive, hence this means that t⁴ must be non-zero. Furthermore, this implies that

v''=0

Now you have a second order differential equation

\frac{d^2 v}{dt^2}=0

Simply integrate twice with respect to t. Do you follow?

 

[efghi]

So the 2nd general solution should be y₂ = t⁶ / 18 ?

 

[Hootenanny]

So the 2nd general solution should be y₂ = t⁶ / 18 ?

How did you get that?

 

[efghi]

integrate twice with respect to t

integral of t⁴v'' = t⁵ / 3
integral of t⁵ / 3 = t⁶ / 18?

 

[Hootenanny]

integrate twice with respect to t

integral of t⁴v'' = t⁵ / 3
integral of t⁵ / 3 = t⁶ / 18?

That's wrong. As I said in my previous post, you need only solve the ODE

v''=0

And then multiply the solution by y₁ to obtain the general solution.

 

[efghi]

How do i solve the ODE?

 

[Hootenanny]

How do i solve the ODE?

As I said before integrate twice with respect to t.