23.3.573 problem 581

Internal problem ID [6287]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 581
Date solved : Tuesday, September 30, 2025 at 02:44:57 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 178

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

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

✓ Mathematica. Time used: 0.789 (sec). Leaf size: 199

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

\begin{align*} y(x)&\to \sqrt {a+x (b+c x)} \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right ) \left (c_1 \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+\frac {c_2}{\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}\right ) \end{align*}

✗ Sympy

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

False