How to DSolve this differential equation?

Question

How can I find h[r] from this equation?
I need h[r] or its r derivative.
(E^(Derivative[1][h][r]^2/(2*b^2))*Sqrt[r^4*Sin[[Theta][]]^2]*
(-(r*Derivative[1][h][r]^2*Derivative[2][h][r]) - 
b^2*(2*Derivative[1][h][r] + r*Derivative[2][h][r])))/
(4*b^2*Pi*r) == 0

In fact, the output of Mmais kind of vague. Is there anything wrong?

Bob Hanlon · Accepted Answer

Clear["Global`*"]

eqn = (E^(h'[r]^2/(2*b^2))*
      Sqrt[r^4*Sin[θ]^2]*(-(r*h'[r]^2*h''[r]) - 
        b^2*(2*h'[r] + r*h''[r])))/(4*b^2*Pi*r) == 0;

DSolve[eqn, h, r]

Verifying the solution,
eqn /. sol // Simplify

(* {True, True} *)

Since h[r] is an integral, it is easier to deal with its derivative
h'[x] /. sol

(* {-b Sqrt[ProductLog[E^((2 C[1])/b^2)/(b^2 x^4)]], 
 b Sqrt[ProductLog[E^((2 C[1])/b^2)/(b^2 x^4)]]} *)

Plotting the derivative,
Plot[Evaluate@
  Table[
   Tooltip[h'[x] /. sol /. C[1] -> 1, b],
   {b, 1/2, 3/2, 1/2}],
 {x, -5, 5},
 PlotStyle -> {Dashed, Automatic},
 PlotLegends -> "Expressions"]

How to DSolve this differential equation?

One Answer

Add your own answers!

Ask a Question