ODE
\[ a e^{b x} y(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0317967 (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 = 0.129 (sec), leaf count = 39
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\sl J}_{0}\left (2\,{\frac {\sqrt {a}{{\rm e}^{1/2\,bx}}}{b}}\right )}+{\it \_C2}\,{{\sl Y}_{0}\left (2\,{\frac {\sqrt {a}{{\rm e}^{1/2\,bx}}}{b}}\right )} \right \} \] 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),'implicit')
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))