Internal problem ID [11110]
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: 22.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a +b \,{\mathrm e}^{\lambda x}+b -3 \lambda \right ) y^{\prime }+a^{2} \lambda \left (-\lambda +b \right ) {\mathrm e}^{2 \lambda x} y=0} \]
✓ Solution by Maple
Time used: 0.406 (sec). Leaf size: 205
dsolve(diff(y(x),x$2)+(a+b*exp(lambda*x)+b-3*lambda)*diff(y(x),x)+a^2*lambda*(b-lambda)*exp(2*lambda*x)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {{\mathrm e}^{x \lambda } \left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right )}{2 \lambda }} \left (\operatorname {KummerU}\left (\frac {\left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right ) \left (-2 \lambda +b +a \right )}{2 \sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, \lambda }, \frac {-2 \lambda +b +a}{\lambda }, \frac {\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{2} +\operatorname {KummerM}\left (\frac {\left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right ) \left (-2 \lambda +b +a \right )}{2 \sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, \lambda }, \frac {-2 \lambda +b +a}{\lambda }, \frac {\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 3.799 (sec). Leaf size: 260
DSolve[y''[x]+(a+b*Exp[\[Lambda]*x]+b-3*\[Lambda])*y'[x]+a^2*\[Lambda]*(b-\[Lambda])*Exp[2*\[Lambda]*x]*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \exp \left (-\frac {e^{\lambda x} \left (\sqrt {-4 a^2 b \lambda +4 a^2 \lambda ^2+b^2}+b\right )}{2 \lambda }\right ) \left (c_1 \operatorname {HypergeometricU}\left (\frac {(a+b-2 \lambda ) \left (b+\sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}\right )}{2 \lambda \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}},\frac {a+b-2 \lambda }{\lambda },\frac {e^{x \lambda } \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}{\lambda }\right )+c_2 L_{-\frac {(a+b-2 \lambda ) \left (b+\sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}\right )}{2 \lambda \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}}^{\frac {a+b-3 \lambda }{\lambda }}\left (\frac {e^{x \lambda } \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}{\lambda }\right )\right ) \]