# Partial differential equation - DSolve gives no output, but solution exists

I have trouble obtaining solution for equation using DSolve:

My mathematica code is given by:

DSolve[\[Kappa]^2 *T + (1/2)*f[G, T] - (1/2) G *D[f[G, T], G] + T* D[f[G, T], T] - (2/(n - 1))*G ^2 D[D[f[G, T], G], G] - 3 n^2 (T/\[Rho]0)^(2/3 n) - (3 n)/(2 (n - 1))*G * T D[D[f[G, T], T], G] == 0, f[G, T], {G, T}]

Using this lines gives me no output (output is exactly my dsolve command) - from what i found on stackexchange it means that exact solution doesn't exist. Hoverver solution for this equation is given by: where $$d_i$$ are integration constants and $$\chi_i$$ are some combinations of $$d$$'s, $$n,\rho_0$$ and $$\kappa$$

How to solve this equation with DSolve command? Is there any mathematical trick to manipulate this equation?

Article [ https://link.springer.com/article/10.1140/epjc/s10052-016-4502-1 ] where this equation is given and solved doesn't gives any additional information about behaviour of function $$f(G,T)$$ or any additional equations - exept that the $$n>0$$

p[t_] := Refine[(po* t^(-3 n)), Assumptions -> n > 0];
G[t_] := Refine[(24*n^3/t^4)*(n - 1), Assumptions -> n > 0];
T[t_] := Refine[(p[t]), Assumptions -> n > 0];

eq = Refine[
k^2*T[t] + (1/2)*f[G[t], T[t]] - (1/2)*G[t]*D[f[G[t], T[t]], G] +
T[t]*D[f[G[t], T[t]], T[t]] - (2/(n - 1))*
G[t]^2 D[f[G[t], T[t]], {G[t], 2}] -
3 n^2 (T[t]/po)^(2/3 n) - (3 n/(2 (n - 1)))*G[t]*T[t] *
D[D[f[G[t], T[t]], {T[t], 1}], {G[t], 1}],
Assumptions -> {k \[Element] Reals, po \[Element] Reals, n > 0,
G[t] \[Element] Reals, T[t] \[Element] Reals,
f[G[t], T[t]] \[Element] Reals, t > 0}]

s1 = DSolve[{eq == 0}, f[G[t], T[t]], {G[t], T[t]}] /. Rule -> Equal

• You cannot solve PDE in this way. I mean, this won't give anything helpful when solving PDE, AFAIK. Just consider the simplest example: {DSolve[3 D[y[x1, x2], x1] + 5 D[y[x1, x2], x2] == x1 , y[x1, x2], {x2}], DSolve[3 D[y[x1, x2], x1] + 5 D[y[x1, x2], x2] == x1 , y[x1, x2], {x1}]}. – xzczd Apr 6 '19 at 16:22
• i looked up the article so i will edit this answer – Alrubaie Apr 6 '19 at 16:30
• you need to notice and define your G & T because they depend on time and they cancel some terms in the PDE, My problem i don't know how to define a function F with variable T&G that depend on function T&G with another variable t time! – Alrubaie Apr 6 '19 at 16:34
• So problem is with definitions? $G,T$ arent independent variables so i have to define them as functions dependent on $t$ and then, i have to solve it w.r.t $T$ and $G$? – StarPlatinumZaWardo Apr 6 '19 at 16:38
• i think so that's it – Alrubaie Apr 6 '19 at 16:40