Internal problem ID [10903]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-3 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 69.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {x y^{\prime \prime }+a x y^{\prime }+a y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 36
dsolve(x*diff(y(x),x$2)+a*x*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y = \left (\operatorname {Ei}_{1}\left (-a x \right ) a x +{\mathrm e}^{a x}\right ) {\mathrm e}^{-a x} c_{1} +{\mathrm e}^{-a x} c_{2} x \]
✓ Solution by Mathematica
Time used: 0.177 (sec). Leaf size: 35
DSolve[x*y''[x]+a*x*y'[x]+a*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-a x} \left (a c_2 x \operatorname {ExpIntegralEi}(a x)-c_2 e^{a x}+c_1 x\right ) \]