# How to solve this differential equation with a square root?

I've to solve this differential equation but I don't know how to do it. Someone can help me?

$$\frac{dy}{dx} = \frac{1}{x} \Bigg( -1 - \frac{2a}{\sqrt{a^2 - 3x L f^2 e^{3y}}} \Bigg)$$

where a, L and f are costant.

I've tried to solve it in Mathematica with DSolve in this way:

DSolve[1/x (-1 - (2 a/Sqrt[a^2 - 3 x L f^2 Exp[3 y[x]]])) == y'[x],
y[x], x]


Thanks everyone.

• Is this question about the Mathematica software? If not, this question is better suited for math.stackexchange. If you are asking about the Mathematica software, could you include what you have tried? – eyorble Mar 30 at 16:19
• yes it's about Mathematica software and I've tried with DSolve – Laria Mar 30 at 16:23
• Please include what you have already tried with DSolve in your question – MarcoB Mar 30 at 16:24
• Your DSolve works in 12.2 without providing values to the constants, and gives multiple solutions of the form y[x]->Log[Root[...]] – flinty Mar 30 at 16:32
• a = 1; f = 1; L = 1; DSolve[ 1/x (-1 - (2 a/Sqrt[a^2 - 3 x L f^2 Exp[3 y[x]]])) == y'[x], y[x], x] produces a useless output. Try ParametricNDSolve. – user64494 Mar 30 at 16:34

Mathematica seems to have problems with Sqrt. "Squaring" gives

ode = ( -(2 a/Sqrt[a^2 - 3 x L f^2 Exp[3 y[x]]]))^2 == (x y'[x] + 1)^2
(*(4 a^2)/(a^2 - 3 E^(3 y[x]) f^2 L x) == (1 + x Derivative[y][x])^2*)


DSolveis now able to solve the ode

DSolve[ode, y[x], x][]
(*Solve[(3 (6 Sqrt[a^2 x^2 (a^2 - 3 E^(3 y[x]) f^2 L x)]
ArcTan[Sqrt[-a^2 + 3 E^(3 y[x]) f^2 L x]/(3 a)] -
2 Sqrt[a^2 x^2 (a^2 - 3 E^(3 y[x]) f^2 L x)]
ArcTan[Sqrt[-a^2 + 3 E^(3 y[x]) f^2 L x]/a] +
a x Sqrt[-a^2 +
3 E^(3 y[x]) f^2 L x] (5 Log[x] +
3 Log[8 a^2 + 3 E^(3 y[x]) f^2 L x])))/(16 a x Sqrt[-a^2 +
3 E^(3 y[x]) f^2 L x]) - (3 y[x])/16 == C, y[x]]*)


As the Maple result, Mathematica evaluates an implicit equation in y[x]!

• You obtain a non-equivalent ODE by squaring. – user64494 Mar 31 at 15:35
• This ode includes all solution branches of the original ode! – Ulrich Neumann Mar 31 at 17:30

As mentioned by flinty in the comment, v12.2 can solve the problem directly, with a Solve::ifun warning generated. If you need the implicit solution as given by Maple, you can:

Trace[
DSolve[1/x (-1 - (2 a/Sqrt[a^2 - 3 x L f^2 Exp[3 y[x]]])) == y'[x], y[x], x],
Solve[_, y[x]], TraceInternal -> True] // Flatten
(*
{HoldForm[
Solve[(3/8)*(3*Log[a - Sqrt[a^2 - 3*E^(3*y[x])*f^2*L*x]] +
2*Log[a + Sqrt[a^2 - 3*E^(3*y[x])*f^2*L*x]] +
3*Log[3*a + Sqrt[a^2 - 3*E^(3*y[x])*f^2*L*x]]) - 3*y[x] == C, y[x]]]}
*)