# (Ordinary) Differential Equation Trouble

• Euler2718
It's hard to tell from just what you have written so far.In summary, the conversation discusses finding the solution to a differential equation using substitution method of a Bernoulli equation. It involves using the substitution v = y^(1-n) and solving for v and v'. However, there is some confusion about the final answer and whether it satisfies the original DE. Further work is needed to determine the correctness of the solution.
Euler2718

## Homework Statement

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

$$t^{2}y^{\prime} + 2ty - y^{3} = 0$$

## Homework Equations

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

## The Attempt at a Solution

[/B]
$$t^{2}y^{\prime} + 2ty - y^{3} = 0$$
$$\implies \frac{t^{2}y^{\prime}}{y^{3}} + \frac{2t}{y^{2}} = 1$$
Let $v = \frac{1}{y^{2}}$, then $v^{\prime} = -\frac{2y^{\prime}}{y^{3}}$

This is where I'm a little lost with respect to the given solution. It tells me that after substituing in $v$ and $v^{\prime}$ I should be getting:
$$t^{2}v^{\prime} - 2tv = -2$$
But I get:
$$-\frac{1}{2}t^{2}v^{\prime} + 2tv =1$$

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.

Morgan Chafe said:

## Homework Statement

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

$$t^{2}y^{\prime} + 2ty - y^{3} = 0$$

## Homework Equations

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

## The Attempt at a Solution

[/B]
$$t^{2}y^{\prime} + 2ty - y^{3} = 0$$
$$\implies \frac{t^{2}y^{\prime}}{y^{3}} + \frac{2t}{y^{2}} = 1$$
Let $v = \frac{1}{y^{2}}$, then $v^{\prime} = -\frac{2y^{\prime}}{y^{3}}$

This is where I'm a little lost with respect to the given solution. It tells me that after substituing in $v$ and $v^{\prime}$ I should be getting:
$$t^{2}v^{\prime} - 2tv = -2$$
But I get:
$$-\frac{1}{2}t^{2}v^{\prime} + 2tv =1$$

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.
Your work looks correct. One way to know for sure is to push it through to a solution, then check whether your solution satisfies the original DE.

LCKurtz said:
Your work looks correct. One way to know for sure is to push it through to a solution, then check whether your solution satisfies the original DE.

Is this :

$$v^{\prime} - \frac{4}{t}v = \frac{2}{t^{2}}$$

Valid? Can I divide through by $t^{2}$ to make it into the form: $y^{\prime}(t) + p(t)y(t) = q(t)$. I still can't seem to get an answer that works if I substitute it back in.

Morgan Chafe said:
But I get:
$$-\frac{1}{2}t^{2}v^{\prime} + 2tv =1$$
Morgan Chafe said:
Is this :

$$v^{\prime} - \frac{4}{t}v = \frac{2}{t^{2}}$$

Valid? Can I divide through by $t^{2}$ to make it into the form: $y^{\prime}(t) + p(t)y(t) = q(t)$. I still can't seem to get an answer that works if I substitute it back in.

Aren't you missing a sign on the right side from what you posted earlier (above)?

LCKurtz said:
Aren't you missing a sign on the right side from what you posted earlier (above)?

Oh, that should be a $-\frac{2}{t^{2}}$. But that was just a LaTeX mistake.

Morgan Chafe said:
Oh, that should be a $-\frac{2}{t^{2}}$. But that was just a LaTeX mistake.
In that case, you need to show the rest of your work before we can help you find what is wrong.

## 1. What are ordinary differential equations (ODEs)?

Ordinary differential equations (ODEs) are mathematical equations that describe the relationship between an unknown function and its derivatives. They are used to model various phenomena in physics, engineering, and other fields.

## 2. What are some common types of ODEs?

Some common types of ODEs include linear first-order equations, separable equations, exact equations, and second-order equations. These types differ in their form and the methods used to solve them.

## 3. How are ODEs solved?

ODEs are solved using various techniques, including separation of variables, substitution, and integration. The specific method used depends on the type of ODE and its initial conditions.

## 4. What are some applications of ODEs?

ODEs have many applications in science and engineering, such as modeling population growth, predicting the motion of planets, and analyzing electrical circuits. They are also used in economics, biology, and other fields.

## 5. What are some challenges in solving ODEs?

Solving ODEs can be challenging due to the complexity of the equations and the need to find the appropriate solution method. Additionally, some ODEs may not have closed-form solutions, requiring numerical methods for approximation.

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