Internal problem ID [11124]
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: 26.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{x}+b \right ) y^{\prime }+\left (c \left (-c +a \right ) {\mathrm e}^{2 x}+\left (a k +b c -2 c k +c \right ) {\mathrm e}^{x}+k \left (b -k \right )\right ) y=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 100
dsolve(diff(y(x),x$2)+(a*exp(x)+b)*diff(y(x),x)+( c*(a-c)*exp(2*x)+ (a*k+b*c+c-2*c*k)*exp(x) + k*(b-k) )*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-k x -{\mathrm e}^{x} c}+c_{2} \left (-{\mathrm e}^{-\frac {a \,{\mathrm e}^{x}}{2}-\frac {\left (b +2\right ) x}{2}} \left (-1+b -2 k \right ) \operatorname {WhittakerM}\left (-\frac {b}{2}+k +1, -\frac {b}{2}+k +\frac {1}{2}, \left (a -2 c \right ) {\mathrm e}^{x}\right )+\left (a -2 c \right ) \operatorname {WhittakerM}\left (-\frac {b}{2}+k , -\frac {b}{2}+k +\frac {1}{2}, \left (a -2 c \right ) {\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {a \,{\mathrm e}^{x}}{2}-\frac {x b}{2}}\right ) \]
✓ Solution by Mathematica
Time used: 3.806 (sec). Leaf size: 71
DSolve[y''[x]+(a*Exp[x]+b)*y'[x]+( c*(a-c)*Exp[2*x]+ (a*k+b*c+c-2*c*k)*Exp[x] + k*(b-k) )*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-c e^x} \left (e^x\right )^{-k} \left (c_1-c_2 \left (e^x\right )^{2 k-b} \left (e^x (a-2 c)\right )^{b-2 k} \Gamma \left (2 k-b,(a-2 c) e^x\right )\right ) \]