Solving a Differential Equation with a Substitution

In summary, the conversation discusses a differential equation and various suggested substitutions to solve it. After trying different options, the speaker finds that the substitution u=2x+y simplifies the equation and leads to a solution.
  • #1
FeDeX_LaTeX
Gold Member
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13
I have the differential equation:

[tex] 4(2x^2 + xy) \frac{dy}{dx} = 3y^2 + 4xy[/tex]

The only thing I could see working is a substitution, but I can't work out which one to use. I've tried letting v = xy, or v = y/x, but neither of those seem to produce anything useful. Can anyone give me a hint?
 
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  • #2
i thought it was supposed to be y=xv... that's for the homogeneous ones i think. i am not expert though i am in the class right now myself.
 
  • #3
You're right, y = vx does work... sorry, idiotic me didn't even check it properly. Thanks!
 
  • #4
Refactor the left hand side to 4x(2x+y)dy/dx. This suggests trying u=2x+y. This will simplify nicely, and should in turn suggest trying v=u2.
 
  • #5
Thanks, that worked out even better -- I think that may have been the intended solution.
 

1. What is a substitution method for solving a differential equation?

The substitution method is a technique used to solve differential equations by substituting a new variable in place of the original variable. This new variable is chosen in such a way that it simplifies the differential equation into an easier form, making it easier to solve.

2. When should I use the substitution method for solving a differential equation?

The substitution method is most useful when the differential equation cannot be solved using other methods such as separation of variables or the method of undetermined coefficients. It is also useful for solving linear differential equations with homogeneous coefficients.

3. How do I choose the substitution variable for solving a differential equation?

The substitution variable should be chosen based on the terms in the differential equation. It should be a variable that when substituted, will eliminate or simplify the highest order derivative in the equation. A commonly used substitution variable is u = y'.

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

No, the substitution method is most effective for solving first order and second order linear differential equations with homogeneous coefficients. It may not work for higher order equations or non-linear equations.

5. What are the steps involved in solving a differential equation with a substitution?

The general steps for solving a differential equation with a substitution are:

  • Identify the substitution variable
  • Substitute the variable into the differential equation
  • Simplify the equation using algebraic manipulation
  • Integrate both sides of the equation
  • Solve for the original variable
  • Check the solution for accuracy

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