Who can find the solution to this ODE?

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In summary, the conversation discusses rewriting a given ODE as a Bernoulli equation and using exact differentials to solve it. The solution is given as y=-1/(x^2+cx), where c is a constant. The conversation also acknowledges the effectiveness of the solution."
  • #1
Cody Palmer
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Looking for the solution to the following ODE:
[tex]
\[
\frac{dy}{dx} = \frac{x^2y^2 - y}{x}
\]
[/tex]
 
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  • #2
rewrite it as

[tex]\frac{dy}{dx}=xy^2-\frac{y}{x} \Rightarrow \frac{dy}{dx}+\frac{y}{x}=xy^2[/tex]


then read about http://en.wikipedia.org/wiki/Bernoulli_differential_equation" [Broken]
 
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  • #3
I had not thought of turning it into a Bernoulli equation. Actually though I found out another, more elementary way of doing it using exact differentials:
[tex]
\[
\frac{dy}{dx} = \frac{x^2y^2 - y}{x} \Rightarrow x dy = x^2y^2 dx - y dx
\]
[/tex]
Divide through by [tex]y^2[/tex]
[tex]
\[
x\frac{1}{y^2} dy = x^2 dx - \frac{1}{y} dx \Rightarrow x\frac{1}{y^2} dy - (- \frac{1}{y}) dx = x^2 dx
\]
\[
\frac{x\frac{1}{y^2} dy - (- \frac{1}{y}) dx}{x^2} = dx
\]
[/tex]
The LHS is the exact differential for the quotient [tex] \frac{-\frac{1}{y}}{x}[/tex]
So we can integrate both sides to get
[tex]
\[
-\frac{1}{xy} = x + c \Rightarrow y=-\frac{1}{x^2 + cx}
\]
[/tex]
 
  • #4
Nice Cody Palmer, very nice :approve:

coomast
 

1. Who is responsible for finding the solution to an ODE?

The responsibility for finding the solution to an ODE (Ordinary Differential Equation) lies with mathematicians and scientists who specialize in the field of differential equations. They use various mathematical techniques and algorithms to find solutions to different types of ODEs.

2. Can anyone find the solution to an ODE?

In theory, anyone with a strong background in mathematics and a good understanding of ODEs can find the solution to a given equation. However, in practice, it requires a lot of knowledge and expertise to solve complex ODEs.

3. Are there any software programs that can find the solution to an ODE?

Yes, there are many software programs specifically designed to solve ODEs, such as MATLAB, Mathematica, and Maple. These programs use sophisticated algorithms to find solutions to both simple and complex ODEs.

4. How long does it take to find the solution to an ODE?

The time it takes to find the solution to an ODE depends on various factors, including the complexity of the equation, the method used to solve it, and the computational power available. For simple ODEs, it may only take a few minutes, while for more complex ones, it may take hours or even days.

5. Is there only one solution to an ODE?

No, there can be multiple solutions to an ODE, depending on the initial conditions and the method used to solve it. Some ODEs may have infinite solutions, while others may have a finite number of solutions. It all depends on the specific equation and its properties.

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