I'd like to see a new function called DChangeTransform[] which can change the variables of differential equations, but there seems to be no such function in Mathematica.
Examples:
DChangeTransform[D[f[x], {x, 2}] - f[x] == 0, {t = Exp[x]}, {x}, {t}, {f[x]}]
(* t^2*f''[t]+t*f'[t]-f[t]=0 *)
DChangeTransform[D[f[x], {x, 2}] - f[x] == 0, {t = Log[x]}, {x}, {t}, {f[x]}]
(* Exp[2*t]*f[t]+f'[t]-f''[t]=0 *)
DChangeTransform[D[f[x], x] - f[x] == 0, {t = Tan[x]}, {x}, {t}, {f[x]}]
(* (1+t^2)*f'[t]+f[t]=0 *)
DChangeTransform[D[u[x, t], {t, 2}] == c^2 D[u[x, t], {x, 2}], {a == x + c t, r == x - c t}, {x, t}, {a, r}, {u[x, t]}]
(* c (u^(1,1))[a,r]=0 *)
DChangeTransform[ D[z[x, y], {x, 2}] - D[z[x, y], {x, 1}, {y, 1}] - 2*D[z[x, y], {y, 2}] == 0, {u == 2*x + y, v == y - x}, {x, y}, {u, v}, {z[x, y]}]
(* -9 (z^(1,1))[u,v]=0 *)
DChangeTransform[x^2*D[f[x, y], {x, 2}] - 2*x*y*D[f[x, y], {x, 1}, {y, 1}] + y^2*D[f[x, y], {y, 2}] + x*D[f[x, y] == 0, {x, 1}] + y*D[f[x, y], {y, 1}], {u == x, v == x*y}, {x, y}, {u, v}, {f[x, y]}]
(* u ((f^(1,0))[u,v]+u (f^(2,0))[u,v])=0 *)
DChangeTransform[x*y^3*D[f[x, y], {x, 2}] + x^3*y*D[f[x, y], {y, 2}] - y^3*D[f[x, y], {x, 1}] - x^3*D[f[x, y], {y, 1}] == 0, {u == y^2, v == x^2}, {x, y}, {u, v}, {f[x, y]}]
(* u^(3/2) v^(3/2) ((f^(0,2))[u,v]+(f^(2,0))[u,v])=0 *)
DChangeTransform[D[f[x, y], x, x] + D[f[x, y], y, y] == 0, "Cartesian" -> "Polar", {x, y}, {r, t}, f[x, y]]
(* (f^(0,2))[r,t]+r ((f^(1,0))[r,t]+r (f^(2,0))[r,t])=0 *)
DChangeTransform[D[f[x, y, z], {x, 2}] + D[f[x, y, z], {y, 2}] + D[f[x, y, z], {z, 2}] == 0, "Cartesian" -> "Spherical", {x, y, z}, {r, t, s}, f[x, y, z]]
(*(Csc[t]^2 (f^(0,0,2))[r,t,s])/r^2+(Cot[t] \(f^(0,1,0))[r,t,s])/r^2+(f^(0,2,0))[r,t,s]/r^2+(2 \(f^(1,0,0))[r,t,s])/r+(f^(2,0,0))[r,t,s]=0 *)
DChangeTransform[(x^2 + y^2)*D[u[x, y], x, x] + D[u[x, y], y, y] == 0, "Cartesian" -> "Polar", {x, y}, {r, t}, u[x, y]]
(* 2 (-1+r^2) Sin[2 t] (u^(0,1))[r,t]+(1+r^2-(-1+r^2) Cos[2 t]) \(u^(0,2))[r,t]+r ((1+r^2-(-1+r^2) Cos[2 t]) (u^(1,0))[r,t]-2 (-1+r^2) \Sin[2 t] (u^(1,1))[r,t]+r (1+r^2+(-1+r^2) Cos[2 t]) (u^(2,0))[r,t])=0 \ *)
DChangeTransform[2 (-1 + r^2) Sin[2 t] D[u[r, t], {t, 1}] + (1 + r^2 - (-1 + r^2) Cos[2 t]) D[ u[r, t], {t, 2}] + r ((1 + r^2 - (-1 + r^2) Cos[2 t]) D[u[r, t], {r, 1}] - 2 (-1 + r^2) Sin[2 t] D[u[r, t], {r, 1}, {t, 1}] + r (1 + r^2 + (-1 + r^2) Cos[2 t]) D[u[r, t], {r, 2}]) == 0, "Polar" -> "Cartesian", {r, t}, {x, y}, u[r, t]]
(* (u^(0,2))[x,y]+(x^2+y^2) (u^(2,0))[x,y]=0 *)