Solving a differential equation with two parameters

Question

How can I code and solve the following differential equation in Mathematica:

$$\frac{\mathrm{d}}{\mathrm{d}x}\left(x^2 e^{2ax}\right) \frac{\mathrm{d}y}{\mathrm{d}x} + b x^2 y=0,$$

where $a$ and $b$ are constants?

Answer 1

Clear["Global`*"]

eqn = D[x^2*E^(2*a*x), x]*y'[x] + b*x^2*y[x] == 0

(* b x^2 y[x] + (2 E^(2 a x) x + 2 a E^(2 a x) x^2) y'[x] == 0 *)

Solving for y as a pure function (facilitates substitution into the original equation)

sol = DSolve[eqn, y, x][[1]]

(* {y -> Function[{x}, 
   E^((b (E^(-2 a x) + 2 E^2 ExpIntegralEi[-2 - 2 a x]))/(4 a^2)) C[1]]} *)

Verifying that the solution satisfies the equation,

eqn /. sol // Simplify

(* True *)

y[x] is then

y[x] /. sol

(* E^((b (E^(-2 a x) + 2 E^2 ExpIntegralEi[-2 - 2 a x]))/(4 a^2)) C[1] *)

where ExpIntegralEi is the exponential integral function and C[1] is an arbitrary constant