61.34.32 problem 32

Internal problem ID [12796]
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
Date solved : Tuesday, January 28, 2025 at 04:20:13 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+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 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 70

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)
 

\[ y = \left (c_{1} \left (\int {\mathrm e}^{\frac {c x \mu \lambda +b \,{\mathrm e}^{\mu x} \lambda +{\mathrm e}^{\lambda x} a \mu }{\mu \lambda }}d x \right )+c_{2} \right ) {\mathrm e}^{\frac {-{\mathrm e}^{\lambda x} a \mu -\lambda \left (c x \mu +b \,{\mathrm e}^{\mu x}\right )}{\mu \lambda }} \]

✓ Solution by Mathematica

Time used: 35.592 (sec). Leaf size: 113

DSolve[D[y[x],{x,2}]+(a*Exp[\[Lambda]*x]+b*Exp[\[Mu]*x]+c)*D[y[x],x]+(a*\[Lambda]*Exp[\[Lambda]*x]+b*\[Mu]*Exp[\[Mu]*x])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} 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 ) \\ y(x)\to c_2 e^{-\frac {a e^{\lambda x}}{\lambda }-\frac {b e^{\mu x}}{\mu }-c x} \\ \end{align*}