61.27.30 problem 40

Internal problem ID [12461]
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 : Wednesday, March 05, 2025 at 07:06:59 PM
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*}

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

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

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

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

ode=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; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.011 (sec). Leaf size: 3

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq((a*x**2 + b*x + c)*Derivative(y(x), x) + (a*b*x**3 + a*c*x**2 + b)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = O\left (1\right ) \]