First order nonlinear ODE - Integrating factor + exact differentials, or not?

[LawlQuals]

First order nonlinear ODE -- Integrating factor + exact differentials, or not?

Hello everyone,

(I apologize if this did not format properly, if not I will attempt to edit it if that functionality is available upon submitting a question).

I recently came across the following nonlinear ODE that I am having difficulty in solving.

Homework Statement

Determine the general solution to the differential equation:

$$\frac{dy}{dx} = \frac{x^2 - y^2}{x^2 + 5yx} = \frac{(x+y)(x-y)}{x(x + 5y)}$$

Homework Equations

Any methods up to and including a typical advanced engineering mathematics (undergrad) course (integrating factors, variation of parameters, transforms, exact differentials, etc.)

The Attempt at a Solution

My only ideas have involved forcing the equation to be an exact differential. However, I have not been able to come to a solution by this route. To demonstrate, I pursued putting the equation in the form of conventional notation for exact differentials,

$$\underbrace{(x^2 - y^2)}_{=\, M}dx - \underbrace{(x^2 + 5yx)}_{=\, -N}dy = 0$$

$$Mdx + Ndy = 0$$

where

$$M = x^2 - y^2 \Rightarrow M_x = 2x, \quad M_y = -2y$$
$$N = -(x^2 + 5yx) \Rightarrow N_x = -2x - 5x \quad N_y = 5x$$

and the subscripts denote differentiation with respect to the labeled parameter. It is evident it is not exact ($$M_y \neq N_x$$). Further, I was unable to implement an integrating factor $$\sigma$$ such that $$\frac{\partial}{\partial y}(\sigma M) = \frac{\partial}{\partial x} (\sigma N)$$ upon multiplication of the entire equation by $$\sigma$$. It can readily be seen that the integrating factor $$\sigma$$ is not only a function of $$x$$ or $$y$$, e.g. writing out the condition for exact differentials involving the factor $$\sigma$$, and enforcing $$\sigma$$ to be a function of $$x$$ or $$y$$ alone involves results of the form

$$\frac{\sigma_y}{\sigma} \sim \frac{M_y - N_x}{M}$$

or

$$\frac{\sigma_x}{\sigma} \sim \frac{M_y - N_x}{N}$$

which demonstrates that $$\sigma$$ is a function of both variables in either case (in contradiction to the posits that lead to these equations).

Choosing a form $$\sigma (x,y) = x^a y^b$$, I could not discern proper constants $$a$$, and $$b$$.

Does anyone have any suggestions regarding solving this equation? Thanks very much for any insight, and take care.

 

[Mark44]

Hello everyone,

(I apologize if this did not format properly, if not I will attempt to edit it if that functionality is available upon submitting a question).

No need for apologies - what you have looks great.

I recently came across the following nonlinear ODE that I am having difficulty in solving.

Homework Statement

Determine the general solution to the differential equation:

$$\frac{dy}{dx} = \frac{x^2 - y^2}{x^2 + 5yx} = \frac{(x+y)(x-y)}{x(x + 5y)}$$

Homework Equations

Any methods up to and including a typical advanced engineering mathematics (undergrad) course (integrating factors, variation of parameters, transforms, exact differentials, etc.)

The Attempt at a Solution

My only ideas have involved forcing the equation to be an exact differential. However, I have not been able to come to a solution by this route. To demonstrate, I pursued putting the equation in the form of conventional notation for exact differentials,

$$\underbrace{(x^2 - y^2)}_{=\, M}dx - \underbrace{(x^2 + 5yx)}_{=\, -N}dy = 0$$

$$Mdx + Ndy = 0$$

where

$$M = x^2 - y^2 \Rightarrow M_x = 2x, \quad M_y = -2y$$
$$N = -(x^2 + 5yx) \Rightarrow N_x = -2x - 5x \quad N_y = 5x$$

and the subscripts denote differentiation with respect to the labeled parameter. It is evident it is not exact ($$M_y \neq N_x$$). Further, I was unable to implement an integrating factor $$\sigma$$ such that $$\frac{\partial}{\partial y}(\sigma M) = \frac{\partial}{\partial x} (\sigma N)$$ upon multiplication of the entire equation by $$\sigma$$. It can readily be seen that the integrating factor $$\sigma$$ is not only a function of $$x$$ or $$y$$, e.g. writing out the condition for exact differentials involving the factor $$\sigma$$, and enforcing $$\sigma$$ to be a function of $$x$$ or $$y$$ alone involves results of the form

$$\frac{\sigma_y}{\sigma} \sim \frac{M_y - N_x}{M}$$

or

$$\frac{\sigma_x}{\sigma} \sim \frac{M_y - N_x}{N}$$

which demonstrates that $$\sigma$$ is a function of both variables in either case (in contradiction to the posits that lead to these equations).

Choosing a form $$\sigma (x,y) = x^a y^b$$, I could not discern proper constants $$a$$, and $$b$$.

Does anyone have any suggestions regarding solving this equation? Thanks very much for any insight, and take care.

Try this substitution: Let v = y/x ==> y = vx ==> y' = v'x + v

Then the DE y' = (x^2 - y^2)/(x^2 + 5xy) becomes
v'x + v = (x^2 - v^2 x)/(x^2 + 5vx^2)
$$v'x + v = \frac{x^2 - v^2x^2}{x^2 + 5vx^2}$$

After a bit of algebraic manipulation, you get a DE in v and x that is separable. Undo the substitution and you have your solution.