Is there any known closed-form solutions to $y''yx+y'y-(y')^2x=x^3$?

Question

The title says it all...

The equation $y''yx+y'y+(y')^2x=x^3$ belongs to a known DE class (listed on EqWorld).

The equation $y''yx+y'y-(y')^2x=x^3$ appears to be way more difficult to solve...

Is the analytic solution to this equation known at all?

Answer 1

I've solved this. There indeed is a closed-form solution that is obtained as follows:

The equation is a generalized homogeneous equation, so we start by substituting $$x \rightarrow kx, \qquad y \rightarrow k^my, \qquad y' \rightarrow k^{m-1}y', \qquad y'' \rightarrow k^{m-2}y''$$ and obtain $m=2$ because $x^3$ becomes $k^3x^3$ on the right side, and all summands on the left side have the common factor of $k^{2m-1}$.

Next, we substitute $x$ for $e^t$, and $y$ for $ze^{mt}$, i.e. $y=ze^{2t}$

Keeping in mind that $t_x'=\frac{1}{x}=e^{-t}$ and $(e^t)_x'=1$, we write $$ \begin{equation} y_x'=z_t'e^{2t}t_x'+2ze^{2t}t_x'=e^{2t}(z_t'+2z)e^{-t}=e^t(z_t'+2z)\\ y_{xx}''=(z_t'+2z)+e^t(z_{tt}''t_x'+2z_t't_x')=z_t'+2z+z_{tt}''+2z_t'=z_{tt}''+3z_t'+2z\\ ze^{3t}(z'+2z)+ze^{3t}(z''+3z'+2z)-e^{3t}(z'^2+4zz'+4z^2)=e^3t\\ z(z'+2z)+z(z''+3z'+2z)-(z'^2+4zz'+4z^2)=1\\ zz''-z'^2=1 \end{equation} $$ The equation $zz''-z'^2=1$ allows for order reduction by subbing $z'=p(z)$. I'll skip the routine, and just borrow the general solution from wolfram: $$ \begin{equation} z_1(t) = \frac{1}{2} e^{-e^{c_1}t-2c_1-e^{c_1}c_2}(e^{2e^{c_1}(c_2 + t)}+e^{2c_1}) \\ z_2(t) = \frac{1}{2} (e^{-e^{c_1}t-2c_1-e^{c_1}c_2}+e^{e^{c_1}t+e^{c_1}c_2}) \end{equation} $$ All that is left is to perform the backward substitutions, $z\rightarrow y e^{-2t}$ and $e^t \rightarrow x$, to get our closed forms: $$ \begin{equation} y(x) = \frac{1}{2}( x^{2-e^{c_1}}e^{-2c_1-c_2e^{c_1}}+ x^{2+e^{c_1}}e^{c_2e^{c_1}} )\\ y(x) = \frac{1}{2}( x^{2+e^{c_1}}e^{-2c_1+c_2e^{c_1}}+ x^{2-e^{c_1}}e^{-c_2e^{c_1}} ) \end{equation} $$ We can rewrite it further if we want: $$ y(x) = \frac{1}{2}( x^{2\pm e^{c_1}}e^{-2c_1\pm c_2e^{c_1}}+ x^{2\mp e^{c_1}}e^{\mp c_2e^{c_1}} ) $$

There may be typos in the solution, but I've verified the obtained closed forms in Mathcad, and they indeed satisfy the initial equation.

Answer 2

There is no closed-form solution to the differential equation mentioned and generally not even a solution in terms of standard functions.

Sampling some initial values for $y$ and $y'$, without loss of generality, one can receive a general idea about the function $y(x)$, where it seems to exist an asymptotic "duality" between them and thus the inability of describing the solution in closed-form terms is (in an essence) derived :

$ \qquad \qquad \qquad \qquad $

The existence of a solution though, it may be simply derived by standard theorems or just in a functional analysis approach.