How to solve this differential equation?

In summary, the conversation discusses trying to solve a differential equation involving variables x and y. The equation is not exact and the attempt to turn it into an exact differential equation has not been successful. The conversation suggests trying to find an integrating factor and using the hint of taking xlny=t to solve the equation. The final answer should be 2x^2 + (2xlny+1)^2 = c.
  • #1
AdrianZ
319
0

Homework Statement


(xy+2xyln^2y+ylny)dx + (2x^2lny + x)dy = 0


The Attempt at a Solution



well, I've tried my best to solve it and I've filled almost two papers trying to solve it with introducing new variables and substituting and then plugging them in but It hasn't gotten solved yet. any ideas would be appreciated.
 
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  • #2
Try to turn it into an exact differential equation?
 
  • #3
I've tried my best to do that already. Do you know any integrating factors or substitutions to turn it into an exact differential equation?
 
  • #4
Divide through by dx?

You would be looking for an integrating factor if it was a first-order linear ODE, I think.
 
  • #5
  • #6
I checked the answers at the end of the book and it gives a hint to take xlny=t. the final answer should be 2x^2 + (2xlny+1)^2 = c and x=0.

any ideas?
 
  • #7
any ideas?
 
  • #8
Is the equation:

[tex]
(x \, y + 2 \, x \, y \, \ln^{2}{
(y)} + y \, \ln{(y)} ) \, dx + (2 x^{2} \, \ln{(y)} +x ) \, dy = 0
[/tex]
 
  • #9
AdrianZ said:
I checked the answers at the end of the book and it gives a hint to take xlny=t. the final answer should be 2x^2 + (2xlny+1)^2 = c

If this is the general solution, then differentiating it, gives:
[tex]
4 x \, dx + 2 (2 \, x \, \ln{y} + 1) \, 2 \left( \ln{y} \, dx + \frac{x}{y} \, dy \right) = 0
[/tex]
and now the arbitrary constant [itex]C[/itex] is gone. Canceling a common factor of 4 and multiplying by [itex]y[/itex] to get rid of the fractions, we get:
[tex]
x \, y \, dx + (2 \, x \, \ln{y} + 1) (y \, \ln{y} \, dx + x \, dy) = 0
[/tex]

Multiply out and collect the terms in front of [itex]dx[/itex] and [itex]d y[/itex] to see if you get the same equation as you quoted.
 

1. What is a differential equation?

A differential equation is an equation that contains derivatives of an unknown function. It describes the relationship between a function and its derivatives.

2. How do I solve a differential equation?

To solve a differential equation, you need to find the function that satisfies the equation. This can be done analytically, using integration and other mathematical techniques, or numerically, using computer algorithms.

3. What are the different types of differential equations?

There are several types of differential equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). Each type has its own methods for solving them.

4. Can differential equations be solved exactly?

Not all differential equations can be solved exactly. In fact, most real-world problems involve differential equations that cannot be solved analytically. In these cases, numerical methods must be used to approximate the solution.

5. What are some common techniques for solving differential equations?

Some common techniques for solving differential equations include separation of variables, variation of parameters, and using integrating factors. Advanced techniques such as Laplace transforms and Fourier series can also be used for more complex problems.

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