##### 4.25.29 $$a e^{b x} y(x)+y''(x)=0$$

ODE
$a e^{b x} y(x)+y''(x)=0$ ODE Classiﬁcation

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 0.172465 (sec), leaf count = 55

$\left \{\left \{y(x)\to c_1 J_0\left (\frac {2 \sqrt {a} \sqrt {e^{b x}}}{b}\right )+2 c_2 Y_0\left (\frac {2 \sqrt {a} \sqrt {e^{b x}}}{b}\right )\right \}\right \}$

Maple
cpu = 0.615 (sec), leaf count = 39

$\left [y \left (x \right ) = \textit {\_C1} \BesselJ \left (0, \frac {2 \sqrt {a}\, {\mathrm e}^{\frac {b x}{2}}}{b}\right )+\textit {\_C2} \BesselY \left (0, \frac {2 \sqrt {a}\, {\mathrm e}^{\frac {b x}{2}}}{b}\right )\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> BesselJ[0, (2*Sqrt[a]*Sqrt[E^(b*x)])/b]*C[1] + 2*BesselY[0, (2*Sqrt[a]
*Sqrt[E^(b*x)])/b]*C[2]}}

Maple raw input

dsolve(diff(diff(y(x),x),x)+a*exp(b*x)*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1*BesselJ(0,2/b*a^(1/2)*exp(1/2*b*x))+_C2*BesselY(0,2/b*a^(1/2)*exp(1/
2*b*x))]