Internal problem ID [8598]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1018.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+a \,{\mathrm e}^{x b} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.023 (sec). Leaf size: 39
dsolve(diff(diff(y(x),x),x)+a*exp(b*x)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \BesselJ \left (0, \frac {2 \sqrt {a}\, {\mathrm e}^{\frac {b x}{2}}}{b}\right )+c_{2} \BesselY \left (0, \frac {2 \sqrt {a}\, {\mathrm e}^{\frac {b x}{2}}}{b}\right ) \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 47
DSolve[a*E^(b*x)*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \, _0\tilde {F}_1\left (;1;-\frac {a e^{b x}}{b^2}\right )+2 c_2 Y_0\left (\frac {2 \sqrt {a} \sqrt {e^{b x}}}{b}\right ) \\ \end{align*}