55.33.28 problem 237

Internal problem ID [14011]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-7
Problem number : 237
Date solved : Thursday, October 02, 2025 at 09:08:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 270

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

\[ y = {\left (\frac {2 a x +\sqrt {-4 a c +b^{2}}+b}{-2 a x +\sqrt {-4 a c +b^{2}}-b}\right )}^{-\frac {b}{2 \sqrt {-4 a c +b^{2}}}} {\left (\frac {-2 a x +\sqrt {-4 a c +b^{2}}-b}{2 a x +\sqrt {-4 a c +b^{2}}+b}\right )}^{-\frac {k}{2 \sqrt {-4 a c +b^{2}}}} \left (c_1 {\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {a \sqrt {\frac {b^{2}-2 b k +k^{2}-4 m}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+c_2 {\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {a \sqrt {\frac {b^{2}-2 b k +k^{2}-4 m}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right ) \]

✓ Mathematica. Time used: 2.092 (sec). Leaf size: 157

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

\begin{align*} y(x)&\to c_1 \exp \left (\frac {\left (-\sqrt {m} \sqrt {\frac {b^2-2 b k+k^2-4 m}{m}}+b-k\right ) \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}\right )+c_2 \exp \left (\frac {\left (\sqrt {m} \sqrt {\frac {b^2-2 b k+k^2-4 m}{m}}+b-k\right ) \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}\right ) \end{align*}

✗ Sympy

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

False