# Second-order non-linear 2-variable partial differential equation

I am trying to gain some information on a non-linear partial differential equation by issuing the following Mathematica command:

DSolve[ f[x, y] - x D[f[x, y], x, y] == 0 , f[x, y] , {x, y} ]


As an answer I just get the same expression. Does that mean that Mathematica cannot come up with any useful information regarding the solution of such equations?

Are there other things I can try? Is anything known about such differential equations?

Uri

P.S. The function Exp[ 2 Sqrt[ y Log[x] ] ] is "almost" a solution of this differential equation.

• this pde is linear. Why do you call it non-linear? Aug 10, 2023 at 6:49
• The second derivative is multiplied by x, which makes it non-linear. Aug 10, 2023 at 6:51
• No. that is wrong. Nonlinearity is on the dependent variable, not the independent variables. The dependent variable enters the pde as linear. Aug 10, 2023 at 6:53
• Thanks for correcting me. Is there anything you can say regarding the queation itself? Aug 10, 2023 at 7:40
• By $x \partial_x \to \partial_z$ it is the wave equation $\partial_{z,y}f =f$. Aug 10, 2023 at 11:37

Mathematica can't solve this directly. One way around is to help it a little by using separation of variables. By assuming $$f(x,y)= X(x) Y(y)$$

f[x_,y_]:=X[x]*Y[y];
pde=f[x,y]-x D[f[x,y],x,y]==0
Expand[pde[[1]]/f[x,y]]==0


The above shows that we have

$$x \frac{X'}{X} \frac{ Y'}{Y} = 1$$ Which is the same as $$x \frac{X'}{X} = \frac{ Y}{Y'}$$ Hence these both are same and equal to some constant, say $$\lambda$$ which is the eigenvalue. $$x \frac{X'}{X} = \frac{ Y}{Y'} = \lambda$$

So we have two ode's to solve

ode1=x*X'[x]/X[x]==λ
ode2=Y[y]/Y'[y]==λ
solX  =  DSolveValue[ode1,X[x],x]
solY  =  DSolveValue[ode2,Y[y],y]/.C[1]->C[2]


Hence the final solution is

solX*solY


The eigenvalue $$\lambda$$ can be determined from boundary conditions.

## Verification

To verify that the above solution is correct

ClearAll["Global*"]
sol=f->Function[{x,y},E^(y/λ) x^λ C[1] C[2]]
pde=f[x,y]-x D[f[x,y],x,y]==0
pde/.sol


• Thanks. That's interesting! As I said in the post, the function $e^{\sqrt{y \ln x}}$ is "close" to being a solution of the equation. I wonder whether there is an exact solution of a similar form. Such a solution, if it exists, is not separable. Is anything known about non-separable solutions? Aug 10, 2023 at 7:51
• The more general solution is Integrate[c[λ] E^(y/λ) x^λ, λ], where c is an arbitrary function of λ`. Aug 11, 2023 at 1:36