ODE
\[ a y'(x)+b e^{k x} y(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.183526 (sec), leaf count = 83
\[\left \{\left \{y(x)\to e^{-\frac {a x}{2}} \left (c_1 \Gamma \left (1-\frac {a}{k}\right ) J_{-\frac {a}{k}}\left (\frac {2 \sqrt {b e^{k x}}}{k}\right )+c_2 \Gamma \left (\frac {a+k}{k}\right ) J_{\frac {a}{k}}\left (\frac {2 \sqrt {b e^{k x}}}{k}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.87 (sec), leaf count = 57
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-\frac {a x}{2}} \BesselJ \left (\frac {a}{k}, \frac {2 \sqrt {b}\, {\mathrm e}^{\frac {k x}{2}}}{k}\right )+\textit {\_C2} \,{\mathrm e}^{-\frac {a x}{2}} \BesselY \left (\frac {a}{k}, \frac {2 \sqrt {b}\, {\mathrm e}^{\frac {k x}{2}}}{k}\right )\right ]\] Mathematica raw input
DSolve[b*E^(k*x)*y[x] + a*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (BesselJ[-(a/k), (2*Sqrt[b*E^(k*x)])/k]*C[1]*Gamma[1 - a/k] + BesselJ[a/k, (2*Sqrt[b*E^(k*x)])/k]*C[2]*Gamma[(a + k)/k])/E^((a*x)/2)}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)+b*exp(k*x)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*exp(-1/2*a*x)*BesselJ(1/k*a,2/k*b^(1/2)*exp(1/2*k*x))+_C2*exp(-1/2*a*x)*BesselY(1/k*a,2/k*b^(1/2)*exp(1/2*k*x))]