Internal problem ID [11130]
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: 32.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }+\left (a \lambda \,{\mathrm e}^{\lambda x}+b \mu \,{\mathrm e}^{\mu x}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 59
dsolve(diff(y(x),x$2)+(a*exp(lambda*x)+b*exp(mu*x)+c)*diff(y(x),x)+(a*lambda*exp(lambda*x)+b*mu*exp(mu*x))*y(x)=0,y(x), singsol=all)
\[ y = \left (c_{1} \left (\int {\mathrm e}^{x c +\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }+\frac {b \,{\mathrm e}^{\mu x}}{\mu }}d x \right )+c_{2} \right ) {\mathrm e}^{-x c -\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }-\frac {b \,{\mathrm e}^{\mu x}}{\mu }} \]
✓ Solution by Mathematica
Time used: 0.146 (sec). Leaf size: 77
DSolve[y''[x]+(a*Exp[\[Lambda]*x]+b*Exp[\[Mu]*x]+c)*y'[x]+(a*\[Lambda]*Exp[\[Lambda]*x]+b*\[Mu]*Exp[\[Mu]*x])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {a e^{\lambda x}}{\lambda }-\frac {b e^{\mu x}}{\mu }-c x} \left (\int _1^xe^{\frac {e^{\lambda K[1]} a}{\lambda }+c K[1]+\frac {b e^{\mu K[1]}}{\mu }} c_1dK[1]+c_2\right ) \]