What is the best substitution for solving this ODE?

In summary, ODE substitution is a technique used in solving ordinary differential equations (ODEs) by substituting a new variable or function in place of the original variable. It is typically used when the original equation cannot be solved by traditional methods and can make solving complicated ODEs much easier and more efficient. However, there may be limitations in finding a suitable substitution or the substitution may lead to a more complex equation. It is important to carefully consider the equation before using ODE substitution.
  • #1
cue928
130
0
So I am trying to figure out what substitution to use for the following ODE substitution:
x^2*y' + 2xy = 5y^3; I initially moved the 2xy to the right but to no avail because when I tried to divide through by x^2 (to clear the left), I struggle to get the y/x format on the right. If the solution is obvious, I apologize, but we get zip examples in class and apparently I'm not good enough to figure this out on my own.
 
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  • #2
Try dividing by x2 and then http://en.wikipedia.org/wiki/Bernoulli_differential_equation" :wink:.
 
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FAQ: What is the best substitution for solving this ODE?

What is ODE substitution?

ODE substitution is a technique used in solving ordinary differential equations (ODEs) by substituting a new variable or function in place of the original variable. This allows for the equation to be rewritten in a simpler form, making it easier to solve.

When is ODE substitution used?

ODE substitution is typically used when the original equation cannot be solved by traditional methods such as separation of variables or integrating factors. It is also useful when the equation contains an independent variable with a high power or a sum of terms.

3. How do you perform ODE substitution?

To perform ODE substitution, the first step is to identify which variable or function can be substituted for the original variable. This is usually done by looking for patterns in the equation or using a change of variables formula. Once the substitution is made, the equation is simplified and can be solved using standard techniques.

4. What are the benefits of using ODE substitution?

ODE substitution can make solving complicated ODEs much easier and more efficient. It allows for the equation to be rewritten in a simpler form, often reducing the order of the equation or eliminating difficult terms. This can save time and effort in finding a solution.

5. Are there any limitations to ODE substitution?

While ODE substitution can be a useful technique, it may not always be possible to find a suitable substitution for the original variable. In some cases, the substitution may also lead to a more complex equation that is difficult to solve. It is important to carefully consider the equation and the available substitutions before using this method.

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