Internal problem ID [10911]
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: 77.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
\[ \boxed {x y^{\prime \prime }+y^{\prime } \left (a x +b \right )=-c x \left (-c \,x^{2}+a x +b +1\right )} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 180
dsolve(x*diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+c*x*(-c*x^2+a*x+b+1)=0,y(x), singsol=all)
\[ y \left (x \right ) = \int -\frac {-c^{2} {\mathrm e}^{-a x} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) b^{3}-3 c^{2} {\mathrm e}^{-a x} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) b^{2}-2 c^{2} {\mathrm e}^{-a x} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) b -{\mathrm e}^{-a x} x^{-b} c_{1} a^{3}-c^{2} x^{2} a^{2}+{\mathrm e}^{-a x} \left (-a x \right )^{-b} c^{2} \Gamma \left (b +3\right )+c x \,a^{3}+c^{2} x b a +2 c^{2} x a -c^{2} b^{2}-3 c^{2} b -2 c^{2}}{a^{3}}d x +c_{2} \]
✓ Solution by Mathematica
Time used: 61.322 (sec). Leaf size: 92
DSolve[x*y''[x]+(a*x+b)*y'[x]+c*x*(-c*x^2+a*x+b+1)==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \int _1^xe^{-a K[1]} K[1]^{-b} \left (\frac {c \left (-\left ((b+1) \Gamma (b+1,-a K[1]) a^2\right )+\Gamma (b+2,-a K[1]) a^2+c \Gamma (b+3,-a K[1])\right ) K[1]^b (-a K[1])^{-b}}{a^3}+c_1\right )dK[1]+c_2 \]