- #1

Euler2718

- 90

- 3

## Homework Statement

Find the solution of the differential equation by using appropriate method:

[tex] t^{2}y^{\prime} + 2ty - y^{3} = 0 [/tex]

## Homework Equations

I'm thinking substitution method of a Bernoulli equation: [itex] v = y^{1-n} [/itex]

## The Attempt at a Solution

[/B]

[tex] t^{2}y^{\prime} + 2ty - y^{3} = 0 [/tex]

[tex]\implies \frac{t^{2}y^{\prime}}{y^{3}} + \frac{2t}{y^{2}} = 1 [/tex]

Let [itex] v = \frac{1}{y^{2}} [/itex], then [itex]v^{\prime} = -\frac{2y^{\prime}}{y^{3}} [/itex]

This is where I'm a little lost with respect to the given solution. It tells me that after substituing in [itex]v[/itex] and [itex]v^{\prime}[/itex] I should be getting:

[tex] t^{2}v^{\prime} - 2tv = -2 [/tex]

But I get:

[tex]-\frac{1}{2}t^{2}v^{\prime} + 2tv =1 [/tex]

Is there some algebra I'm messing up (or flat out not seeing)? Or is this not the best way to approach the problem? This is an elementary level ODE course so it shouldn't require anything too advance.