Solve the given differential equation

[chwala]

Homework Statement

$(2xy+\cos y+\dfrac{2y}{x}-\sin x)dx +(x^2-y\sin y+1)dy=0$

Relevant Equations

hybrid ode

Ok, i just came up with this ode. How does one go about to solve it?

My attempt

Let $M(x,y) =(2xy+\cos y+\dfrac{2y}{x}-\sin x)$ and $N(x,y) = (x^2-y\sin y+1)$

$\dfrac{\partial M}{\partial y}= 2x - \sin y + \dfrac{2}{x}$ and $\dfrac{\partial N}{\partial x}= 2x$

Not exact.

 

[anuttarasammyak]

Say we have a point $$(x,y)=(x_0,y_0)$$ on the solution, $$(x_0+\Delta x, y_0+\Delta y)$$
is also on the solution, where
$$\frac{\Delta y}{\Delta x}=\frac{dy}{dx}|_{x=x_0,y=y_0}=-\frac{2xy+\cos y+\frac{2y}{x}-\sin x}{x^2-y\sin y+1}|_{x=x_0,y=y_0}$$
In this way we can find successive solution points numerically. I have no idea how to get analytic solution if there is.

 

[Mark44]

Ok, i just came up with this ode.

There's no guarantee that an ODE devised more-or-less at random will have an analytical solution.

How does one go about to solve it?

Your approach of checking for exactness is a good start, but as you found, the equation isn't exact. Many textbooks on differential equations take this approach farther by making additional assumptions about the possible solutions.
@anuttarasammyak's suggestion of going for a numerical solution might be the only way forward here.

 

[chwala]

Thanks for the thoughtful input — actually, I’ve taken this as a personal challenge! I’ll be applying the Runge-Kutta method to this problem and working through it manually as far as possible. I’m also committed to generating more of these hybrid-style problems that push the boundaries of how we see mathematics, rather than sticking to traditional textbook paths.

Your suggestion of checking for exactness was solid, and even though it didn’t yield a direct solution here, it shows the right instinct. I completely agree with @anuttarasammyak — numerical methods may indeed be our best route for a meaningful solution. Let’s stretch the limits a little!