Internal problem ID [10916]
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: 82.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {x y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (2 a x +b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 50
dsolve(x*diff(y(x),x$2)+(a*x^2+b*x+c)*diff(y(x),x)+(2*a*x+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{1} \left (\int \frac {x^{c} {\mathrm e}^{\frac {a \,x^{2}}{2}} {\mathrm e}^{b x}}{x^{2}}d x \right )+c_{2} \right ) {\mathrm e}^{-\frac {a \,x^{2}}{2}} {\mathrm e}^{-b x} x^{-c} x \]
✓ Solution by Mathematica
Time used: 1.772 (sec). Leaf size: 63
DSolve[x*y''[x]+(a*x^2+b*x+c)*y'[x]+(2*a*x+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^{1-c} e^{-\frac {1}{2} x (a x+2 b)} \left (c_2 \int _1^xe^{\frac {1}{2} a K[1]^2+b K[1]} K[1]^{c-2}dK[1]+c_1\right ) \]