Solution to the differential equation $\left(x \csc\left(\frac{y}{x}\right)-y\right) dx + xdy$?

What is the general solution to the differential equation $\left(x \csc\left(\frac{y}{x}\right)-y\right) dx + xdy$?

My work

If Im going to reaarange the differential equation above, it would look like this:

$$x \csc\left(\frac{y}{x}\right) dx-ydx + xdy$$ $$x \csc\left(\frac{y}{x}\right) dx-(ydx - xdy)$$

I got an idea how to deal with the expression $(ydx - xdy)$, but I don't know how to approach the expression $x \csc\left(\frac{y}{x}\right) dx$ to get the general solution of the d.e.

How to get the general solution to the differential equation above?

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I guess the differential equation you are trying to solve is :

$$\left(x \csc\left(\frac{y}{x}\right)-y\right) dx + xdy =0$$

Let $y(x) = xv(x)$, which gives : $\frac{dy(x)}{dx} = v(x) + \frac{dv(x)}{dx}$ ::

$$x\csc(v(x)) + x \bigg(x\frac{dv(x)}{dx} + v(x) \bigg) - xv(x) = 0$$

$$\Leftrightarrow$$

$$x\bigg(\csc(v(x)) + x\frac{dv(x)}{dx}\bigg) = 0$$

$$\Leftrightarrow$$

$$\frac{dv(x)}{dx} = - \frac{\csc(v(x))}{x}$$

$$\Leftrightarrow$$

$$\frac{dv(x)}{dx}\sin(v(x)) = -\frac{1}{x}$$

$$\Rightarrow$$

$$\int \frac{dv(x)}{dx}\sin(v(x))dx = \int-\frac{1}{x}dx$$

$$\Leftrightarrow$$

$$\cos(v(x)) = -\ln(x) + c_1$$

Solving for $v(x)$ gives you :

$$v(x) = \begin{cases} -\arccos(\ln(x) - c_1) \\ \arccos(\ln(x) - c_1)\end{cases}$$

Substitute in the initial $y(x) = xv(x)$ that we used to solve the equation and you will get $y(x)$ !

• Huh we have: $$\int\frac{1}{x}\space\text{d}x=\ln\left|x\right|+\text{C}$$ Nov 15 '17 at 11:49
• The general solution I got from your answer is $y = x\cos^{-1}(ln(x) + c)$. Is that right? Nov 15 '17 at 12:46
• @PalautotKa You should get one with a minus sign $-$ as well, since I gave you 2 cases ! But yes, that one is correct ! Nov 15 '17 at 13:08
• The solutions are : $$y(x) = \begin{cases} x\cos^{-1}(\ln(x) + c_1) \\ -x\cos^{-1}(\ln(x) + c_1) \end{cases}$$ Also, usual functions (for example $\cos, \ln$) are written using the command : \ln for $\ln$ for example. Just a tip ! Nov 15 '17 at 13:13
• I wonder why there were 2 solutions........:-) Nov 16 '17 at 8:13

Well, we have:

$$\left(x\cdot\csc\left(\frac{\text{y}}{x}\right)-\text{y}\right)\space\text{d}x+x\space\text{d}\text{y}=0\space\Longleftrightarrow\space\text{y}\space'\left(x\right)=\frac{\text{y}\left(x\right)}{x}-\csc\left(\frac{\text{y}}{x}\right)\tag1$$

Let $\text{y}\left(x\right):=x\cdot\text{v}\left(x\right)$:

$$x\cdot\left(\csc\left(\text{v}\left(x\right)\right)+x\cdot\text{v}\space'\left(x\right)\right)=0\space\Longleftrightarrow\space-\frac{\text{v}\space'\left(x\right)}{\csc\left(\text{v}\left(x\right)\right)}=\frac{1}{x}\tag2$$

Let $\text{u}:=\text{v}\left(x\right)$:

$$\int-\frac{\text{v}\space'\left(x\right)}{\csc\left(\text{v}\left(x\right)\right)}\space\text{d}x=\int-\frac{1}{\csc\left(\text{u}\right)}\space\text{d}\text{u}=\int\frac{1}{x}\space\text{d}x=\ln\left|x\right|+\text{C}\tag3$$

• The general solution I got from your answer is $y = x\cos^{-1}(ln(x) + c)$. Is that right? Nov 15 '17 at 12:54

Substitute $y=xu$ like in a homogeneous ODE to find $$0=\frac{dx}x+\sin(u)du.$$

We need to reaarange the differential equation so that it can be solved easily...

$$\left( x \csc\left( \frac{y}{x}\right) - y\right) dx + x dy = 0$$ $$x \csc\left( \frac{y}{x}\right)dx - ydx + xdy = 0$$ $$xdy = -x \csc\left( \frac{y}{x}\right)dx + ydx$$

Divide the terms of the differential equation by $x$, getting:

$$dy = - \csc\left( \frac{y}{x}\right)dx + \left( \frac{y}{x}\right)dx$$

Apparently this modified form of differential equation is a homogenous differential equation. So we let $v = \frac{y}{x}$, $y = vx$ and $dy = vdx + xdv$. So we get:

$$dy = - \csc\left( \frac{y}{x}\right)dx + \left( \frac{y}{x}\right)dx$$ $$vdx + xdv = - \csc(v)dx + vdx$$ $$xdv = - \csc(v)dx$$ $$\frac{dv}{-\csc(v)} = \frac{1}{x} dx$$ $$-sin(v)dv = \frac{1}{x} dx$$

The above differential equation is now a variable-separable type...so we get the integral of individual terms.

$$-sin(v)dv = \frac{1}{x} dx$$ $$\int -sin(v)dv = \int \frac{1}{x} dx$$ $$cos(v) + c_1 = ln(x) + c_2$$ $$cos(v) = ln(x) + c_2 - c_1$$ $$cos(v) = ln(x) + C$$

Now getting the $v$:

$$cos(v) = ln(x) + C$$ $$v = cos^{-1}(ln(x) + C)$$

The solution of the given differential equation $\left( x \csc\left( \frac{y}{x}\right) - y\right) dx + x dy = 0$ would be:

$$y = vx$$ $$y = (cos^{-1}(ln(x) + C))x$$ $$y = xcos^{-1}(ln(x) + C)$$

Alternate ways of answering it are encouraged....