Internal problem ID [10912]
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: 88.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (a \,x^{3}+b \,x^{2}+2\right ) y^{\prime }+b x y=0} \]
✓ Solution by Maple
Time used: 0.281 (sec). Leaf size: 143
dsolve(x*diff(y(x),x$2)+(a*x^3+b*x^2+2)*diff(y(x),x)+b*x*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{-\frac {\operatorname {csgn}\left (a \right ) x^{2} \left (\operatorname {csgn}\left (a \right )+1\right ) \left (a x +\frac {3 b}{2}\right )}{6}} \operatorname {HeunT}\left (\frac {3^{\frac {2}{3}} b}{2 \left (a^{2}\right )^{\frac {1}{3}}}, -6 \,\operatorname {csgn}\left (a \right ), -\frac {b^{2} 3^{\frac {1}{3}}}{4 \left (a^{2}\right )^{\frac {2}{3}}}, \frac {3^{\frac {2}{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right )+c_{2} {\mathrm e}^{-\frac {\operatorname {csgn}\left (a \right ) x^{2} \left (\operatorname {csgn}\left (a \right )-1\right ) \left (a x +\frac {3 b}{2}\right )}{6}} \operatorname {HeunT}\left (\frac {3^{\frac {2}{3}} b}{2 \left (a^{2}\right )^{\frac {1}{3}}}, 6 \,\operatorname {csgn}\left (a \right ), -\frac {b^{2} 3^{\frac {1}{3}}}{4 \left (a^{2}\right )^{\frac {2}{3}}}, -\frac {3^{\frac {2}{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right )}{x} \]
✓ Solution by Mathematica
Time used: 1.962 (sec). Leaf size: 58
DSolve[x*y''[x]+(a*x^3+b*x^2+2)*y'[x]+b*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {(a x+b) \left (c_2 \int _1^x\frac {e^{-\frac {1}{6} K[1]^2 (3 b+2 a K[1])}}{(b+a K[1])^2}dK[1]+c_1\right )}{b x} \]