61.27.30 problem 40

Internal problem ID [12540]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 40
Date solved : Tuesday, January 28, 2025 at 03:21:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (a b \,x^{3}+a c \,x^{2}+b \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.495 (sec). Leaf size: 165

dsolve(diff(y(x),x$2)+(a*x^2+b*x+c)*diff(y(x),x)+(a*b*x^3+a*c*x^2+b)*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} {\mathrm e}^{-\frac {\left (\left (a \,x^{2}+\frac {3}{2} b x +3 c \right ) \operatorname {csgn}\left (a \right )+a \,x^{2}-\frac {3 b x}{2}-3 c \right ) x \,\operatorname {csgn}\left (a \right )}{6}} \operatorname {HeunT}\left (0, -3 \,\operatorname {csgn}\left (a \right ), -\frac {3^{{1}/{3}} \left (4 a c +b^{2}\right )}{4 \left (a^{2}\right )^{{2}/{3}}}, \frac {3^{{2}/{3}} a \left (2 a x -b \right )}{6 \left (a^{2}\right )^{{5}/{6}}}\right )+c_{2} {\mathrm e}^{-\frac {x \,\operatorname {csgn}\left (a \right ) \left (\left (a \,x^{2}+\frac {3}{2} b x +3 c \right ) \operatorname {csgn}\left (a \right )-a \,x^{2}+\frac {3 b x}{2}+3 c \right )}{6}} \operatorname {HeunT}\left (0, 3 \,\operatorname {csgn}\left (a \right ), -\frac {3^{{1}/{3}} \left (4 a c +b^{2}\right )}{4 \left (a^{2}\right )^{{2}/{3}}}, -\frac {3^{{2}/{3}} a \left (a x -\frac {b}{2}\right )}{3 \left (a^{2}\right )^{{5}/{6}}}\right ) \]

✓ Solution by Mathematica

Time used: 0.787 (sec). Leaf size: 57

DSolve[D[y[x],{x,2}]+(a*x^2+b*x+c)*D[y[x],x]+(a*b*x^3+a*c*x^2+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-\frac {1}{2} x (b x+2 c)} \left (c_2 \int _1^x\exp \left (\frac {1}{6} K[1] (6 c+K[1] (3 b-2 a K[1]))\right )dK[1]+c_1\right ) \]