Internal problem ID [11102]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.3-1. Equations with
exponential functions
Problem number: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-\left (a^{2} {\mathrm e}^{2 x}+a \left (2 b +1\right ) {\mathrm e}^{x}+b^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 71
dsolve(diff(y(x),x$2)-(a^2*exp(2*x)+a*(2*b+1)*exp(x)+b^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{x b +a \,{\mathrm e}^{x}}+c_{2} \left (a^{-2 b -1} \left (-\frac {1}{2}+b \right ) {\mathrm e}^{-x} \operatorname {WhittakerM}\left (-b +1, -b +\frac {1}{2}, 2 a \,{\mathrm e}^{x}\right )-\operatorname {WhittakerM}\left (-b , -b +\frac {1}{2}, 2 a \,{\mathrm e}^{x}\right ) a^{-2 b}\right ) \]
✓ Solution by Mathematica
Time used: 1.8 (sec). Leaf size: 57
DSolve[y''[x]-(a^2*Exp[2*x]+a*(2*b+1)*Exp[x]+b^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{a e^x} \left (e^x\right )^{-b} \left (c_1 \left (e^x\right )^{2 b}-4^b c_2 \left (a e^x\right )^{2 b} \Gamma \left (-2 b,2 a e^x\right )\right ) \]