ODE
\[ b e^{2 a x} y(x)+a y'(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0349082 (sec), leaf count = 78
\[\left \{\left \{y(x)\to \frac {\sqrt {a} e^{-\frac {a x}{2}} \left (c_2 \sin \left (\frac {\sqrt {b e^{2 a x}}}{a}\right )+2 c_1 \cos \left (\frac {\sqrt {b e^{2 a x}}}{a}\right )\right )}{\sqrt {2} \sqrt [4]{b e^{2 a x}}}\right \}\right \}\]
Maple ✓
cpu = 0.133 (sec), leaf count = 39
\[ \left \{ y \left ( x \right ) ={{\rm e}^{-ax}} \left ( \sin \left ( {\frac {{{\rm e}^{ax}}}{a}\sqrt {b}} \right ) {\it \_C1}+\cos \left ( {\frac {{{\rm e}^{ax}}}{a}\sqrt {b}} \right ) {\it \_C2} \right ) \right \} \] Mathematica raw input
DSolve[b*E^(2*a*x)*y[x] + a*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (Sqrt[a]*(2*C[1]*Cos[Sqrt[b*E^(2*a*x)]/a] + C[2]*Sin[Sqrt[b*E^(2*a*x)]/a]))/(Sqrt[2]*E^((a*x)/2)*(b*E^(2*a*x))^(1/4))}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)+b*exp(2*a*x)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = exp(-a*x)*(sin(1/a*b^(1/2)*exp(a*x))*_C1+cos(1/a*b^(1/2)*exp(a*x))*_C2)