23.4.270 problem 270
Internal
problem
ID
[6572]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
4.
THE
NONLINEAR
EQUATION
OF
SECOND
ORDER,
page
380
Problem
number
:
270
Date
solved
:
Friday, October 03, 2025 at 02:09:32 AM
CAS
classification
:
[NONE]
\begin{align*} A y+\left (a +2 b x +c \,x^{2}+y^{2}\right )^{2} y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.045 (sec). Leaf size: 336
ode:=A*y(x)+(a+2*b*x+c*x^2+y(x)^2)^2*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \operatorname {RootOf}\left (c \int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2}+1\right ) \left (-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+c_1 \,\textit {\_f}^{2} c^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+c_1 \,c^{2}+A \right )}}{-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+c_1 \,\textit {\_f}^{2} c^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+c_1 \,c^{2}+A}d \textit {\_f} \sqrt {a c -b^{2}}-c \arctan \left (\frac {c x +b}{\sqrt {a c -b^{2}}}\right )+c_2 \sqrt {a c -b^{2}}\right ) \sqrt {c \,x^{2}+2 b x +a} \\
y &= \operatorname {RootOf}\left (-c \int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2}+1\right ) \left (-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+c_1 \,\textit {\_f}^{2} c^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+c_1 \,c^{2}+A \right )}}{-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+c_1 \,\textit {\_f}^{2} c^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+c_1 \,c^{2}+A}d \textit {\_f} \sqrt {a c -b^{2}}-c \arctan \left (\frac {c x +b}{\sqrt {a c -b^{2}}}\right )+c_2 \sqrt {a c -b^{2}}\right ) \sqrt {c \,x^{2}+2 b x +a} \\
\end{align*}
✓ Mathematica. Time used: 34.874 (sec). Leaf size: 260
ode=A*y[x] + (a + 2*b*x + c*x^2 + y[x]^2)^2*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solve}\left [c \arctan \left (\frac {b+c x}{\sqrt {a c-b^2}}\right )+\sqrt {a c-b^2} \int _1^{\frac {y(x)}{\sqrt {a+x (2 b+c x)}}}\frac {c \left (K[2]^2+1\right )}{\sqrt {\left (K[2]^2+1\right ) \left (A+\left (K[2]^2+1\right ) \left (c_1 c^2+\left (b^2-a c\right ) K[2]^2\right )\right )}}dK[2]=c_2 \sqrt {a c-b^2},y(x)\right ]\\ \text {Solve}\left [c \arctan \left (\frac {b+c x}{\sqrt {a c-b^2}}\right )-\sqrt {a c-b^2} \int _1^{\frac {y(x)}{\sqrt {a+x (2 b+c x)}}}\frac {c \left (K[3]^2+1\right )}{\sqrt {\left (K[3]^2+1\right ) \left (A+\left (K[3]^2+1\right ) \left (c_1 c^2+\left (b^2-a c\right ) K[3]^2\right )\right )}}dK[3]=c_2 \sqrt {a c-b^2},y(x)\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 + 2*b*x + c*x**2 + y(x)**2)**2*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : solve: Cannot solve A*y(x) + (a + 2*b*x + c*x**2 + y(x)**2)**2*Derivative(y(x)