Differential Eq using Substitution help

In summary, the conversation discusses how to solve a differential equation involving the expression (x^2)y' + 2xy = 5y^4. The speaker suggests using the substitution v=1/y^3 and mentions that this equation is of the Bernoulli type. They also encourage the listener to research Bernoulli differential equations for further understanding.
  • #1
g.sharm89
3
0
Hey. I am having a hard time solving this problem.


(x^2)y' + 2xy = 5y^4

I get as far as simplifying to
y' = [(5y^4)/(x^2)] - 2y/x

Then use v: y/x and y: vx & y': v'x + v

And get

v'x + v = [5(v^4)(x^2)] - 2v


And then I get lost. Any help would be appreciated. Thanks!
 
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  • #2
The convenient substitution is v=1/y^3
 
  • #3
Agreed. while [tex]v = y/x[/tex] does work quite frequently (at least in your homework problems I imagine), you have probably learned about other trademark substitutions of different types of equations. Specifically, this is a Bernoulli type equation (and, it has its own characteristic substitution, and JJacquelin has informed which specific substitution to use in this problem). Learning to diagnose a differential equation and determine what type it is, is a quick way to know immediately how to solve them. I encourage you to look up Bernoulli differential equations online, you will find a lot of material.
 

1. What is the concept of substitution in differential equations?

Substitution is a mathematical technique used to simplify complex expressions or equations by replacing variables with simpler expressions or constants. In the context of differential equations, substitution involves replacing the dependent variable with another variable to make the equation easier to solve.

2. When is substitution used in solving differential equations?

Substitution is commonly used in solving differential equations when the equation cannot be solved using direct integration or separation of variables. It is particularly useful for equations that involve complex or nested functions.

3. What are the steps for solving a differential equation using substitution?

The general steps for solving a differential equation using substitution are as follows: 1. Identify the dependent and independent variables in the equation.2. Substitute a new variable for the dependent variable.3. Rewrite the equation in terms of the new variable.4. Solve the resulting equation for the new variable.5. Substitute the original dependent variable back into the solution to find the general solution.

4. Can substitution be used for all types of differential equations?

No, substitution is not always applicable to all types of differential equations. It is most commonly used for first-order ordinary differential equations, but it may also be used for some second-order equations. However, it cannot be used for partial differential equations.

5. Are there any limitations or challenges when using substitution for solving differential equations?

One of the main limitations of substitution is that it may not always result in a simpler or more manageable equation. In some cases, it may even make the equation more complicated. Additionally, substitution may not always lead to a closed-form solution, and numerical methods may be needed to approximate the solution.

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