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 = 0.0372947 (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 = 0.089 (sec), leaf count = 53
\[ \left \{ y \left ( x \right ) ={{\rm e}^{-{\frac {ax}{2}}}} \left ( {{\sl Y}_{{\frac {a}{k}}}\left (2\,{\frac {\sqrt {b}{{\rm e}^{1/2\,kx}}}{k}}\right )}{\it \_C2}+{{\sl J}_{{\frac {a}{k}}}\left (2\,{\frac {\sqrt {b}{{\rm e}^{1/2\,kx}}}{k}}\right )}{\it \_C1} \right ) \right \} \] 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),'implicit')
Maple raw output
y(x) = exp(-1/2*a*x)*(BesselY(1/k*a,2/k*b^(1/2)*exp(1/2*k*x))*_C2+BesselJ(1/k*a,2/k*b^(1/2)*exp(1/2*k*x))*_C1)