23.4.264 problem 264
Internal
problem
ID
[6566]
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
:
264
Date
solved
:
Tuesday, September 30, 2025 at 03:04:35 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} \left (\left (a -y\right ) \left (b -y\right )+\left (a -y\right ) \left (c -y\right )+\left (b -y\right ) \left (c -y\right )\right ) {y^{\prime }}^{2}+2 \left (a -y\right ) \left (b -y\right ) \left (c -y\right ) y^{\prime \prime }&=\operatorname {a3} \left (a -y\right )^{2} \left (b -y\right )^{2}+2 \operatorname {a2} \left (a -y\right )^{2} \left (c -y\right )^{2}+\operatorname {a1} \left (b -y\right )^{2} \left (c -y\right )^{2}+\operatorname {a0} \left (a -y\right )^{2} \left (b -y\right )^{2} \left (c -y\right )^{2} \end{align*}
✓ Maple. Time used: 0.064 (sec). Leaf size: 357
ode:=((b-y(x))*(a-y(x))+(a-y(x))*(c-y(x))+(b-y(x))*(c-y(x)))*diff(y(x),x)^2+2*(a-y(x))*(b-y(x))*(c-y(x))*diff(diff(y(x),x),x) = a3*(a-y(x))^2*(b-y(x))^2+2*a2*(a-y(x))^2*(c-y(x))^2+a1*(b-y(x))^2*(c-y(x))^2+a0*(a-y(x))^2*(b-y(x))^2*(c-y(x))^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\int _{}^{y}\frac {1}{\sqrt {-\operatorname {a0} \,\textit {\_a}^{4}+\textit {\_a}^{3} a \operatorname {a0} +\textit {\_a}^{3} \operatorname {a0} b +\textit {\_a}^{3} \operatorname {a0} c -\textit {\_a}^{2} a \operatorname {a0} b -\textit {\_a}^{2} a \operatorname {a0} c -\textit {\_a}^{2} \operatorname {a0} b c +\textit {\_a} a \operatorname {a0} b c +c_1 \,\textit {\_a}^{3}-c_1 \,\textit {\_a}^{2} a -c_1 \,\textit {\_a}^{2} b -c_1 \,\textit {\_a}^{2} c +c_1 \textit {\_a} a b +c_1 \textit {\_a} a c +c_1 \textit {\_a} b c -c_1 a b c +\textit {\_a}^{2} \operatorname {a1} +2 \textit {\_a}^{2} \operatorname {a2} +\textit {\_a}^{2} \operatorname {a3} -2 \textit {\_a} a \operatorname {a2} -\textit {\_a} a \operatorname {a3} -\textit {\_a} \operatorname {a1} b -\textit {\_a} \operatorname {a1} c -2 \textit {\_a} \operatorname {a2} c -\textit {\_a} \operatorname {a3} b +2 a \operatorname {a2} c +a \operatorname {a3} b +b c \operatorname {a1}}}d \textit {\_a} -x -c_2 &= 0 \\
-\int _{}^{y}\frac {1}{\sqrt {-\operatorname {a0} \,\textit {\_a}^{4}+\textit {\_a}^{3} a \operatorname {a0} +\textit {\_a}^{3} \operatorname {a0} b +\textit {\_a}^{3} \operatorname {a0} c -\textit {\_a}^{2} a \operatorname {a0} b -\textit {\_a}^{2} a \operatorname {a0} c -\textit {\_a}^{2} \operatorname {a0} b c +\textit {\_a} a \operatorname {a0} b c +c_1 \,\textit {\_a}^{3}-c_1 \,\textit {\_a}^{2} a -c_1 \,\textit {\_a}^{2} b -c_1 \,\textit {\_a}^{2} c +c_1 \textit {\_a} a b +c_1 \textit {\_a} a c +c_1 \textit {\_a} b c -c_1 a b c +\textit {\_a}^{2} \operatorname {a1} +2 \textit {\_a}^{2} \operatorname {a2} +\textit {\_a}^{2} \operatorname {a3} -2 \textit {\_a} a \operatorname {a2} -\textit {\_a} a \operatorname {a3} -\textit {\_a} \operatorname {a1} b -\textit {\_a} \operatorname {a1} c -2 \textit {\_a} \operatorname {a2} c -\textit {\_a} \operatorname {a3} b +2 a \operatorname {a2} c +a \operatorname {a3} b +b c \operatorname {a1}}}d \textit {\_a} -x -c_2 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 30.682 (sec). Leaf size: 10537
ode=((a - y[x])*(b - y[x]) + (a - y[x])*(c - y[x]) + (b - y[x])*(c - y[x]))*D[y[x],x]^2 + 2*(a - y[x])*(b - y[x])*(c - y[x])*D[y[x],{x,2}] == a3*(a - y[x])^2*(b - y[x])^2 + 2*a2*(a - y[x])^2*(c - y[x])^2 + a1*(b - y[x])^2*(c - y[x])^2 + a0*(a - y[x])^2*(b - y[x])^2*(c - y[x])^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Too large to display
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
a0 = symbols("a0")
a1 = symbols("a1")
a2 = symbols("a2")
a3 = symbols("a3")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(-a0*(a - y(x))**2*(b - y(x))**2*(c - y(x))**2 - a1*(b - y(x))**2*(c - y(x))**2 - 2*a2*(a - y(x))**2*(c - y(x))**2 - a3*(a - y(x))**2*(b - y(x))**2 + (2*a - 2*y(x))*(b - y(x))*(c - y(x))*Derivative(y(x), (x, 2)) + ((a - y(x))*(b - y(x)) + (a - y(x))*(c - y(x)) + (b - y(x))*(c - y(x)))*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt((a**2*a0*b**2*c**2 - 2*a**2*a0*b**2*c*y(x) + a**2*a0*b**2*