23.4.132 problem 132
Internal
problem
ID
[6434]
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
:
132
Date
solved
:
Tuesday, September 30, 2025 at 02:56:47 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} y y^{\prime \prime }&=\operatorname {a0} +\operatorname {a1} y+\operatorname {a2} y^{2}+\operatorname {a3} y^{3}+\operatorname {a4} y^{4}+{y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 99
ode:=y(x)*diff(diff(y(x),x),x) = a0+a1*y(x)+a2*y(x)^2+a3*y(x)^3+a4*y(x)^4+diff(y(x),x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\int _{}^{y}\frac {1}{\sqrt {\operatorname {a4} \,\textit {\_a}^{4}+2 \ln \left (\textit {\_a} \right ) \textit {\_a}^{2} \operatorname {a2} +2 \operatorname {a3} \,\textit {\_a}^{3}+c_1 \,\textit {\_a}^{2}-2 \operatorname {a1} \textit {\_a} -\operatorname {a0}}}d \textit {\_a} -x -c_2 &= 0 \\
-\int _{}^{y}\frac {1}{\sqrt {\operatorname {a4} \,\textit {\_a}^{4}+2 \ln \left (\textit {\_a} \right ) \textit {\_a}^{2} \operatorname {a2} +2 \operatorname {a3} \,\textit {\_a}^{3}+c_1 \,\textit {\_a}^{2}-2 \operatorname {a1} \textit {\_a} -\operatorname {a0}}}d \textit {\_a} -x -c_2 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 2.383 (sec). Leaf size: 377
ode=y[x]*D[y[x],{x,2}] == a0 + a1*y[x] + a2*y[x]^2 + a3*y[x]^3 + a4*y[x]^4 + D[y[x],x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{\sqrt {\text {a4} K[1]^4+2 \text {a3} K[1]^3+c_1 K[1]^2+2 \text {a2} \log (K[1]) K[1]^2-2 \text {a1} K[1]-\text {a0}}}dK[1]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {\text {a4} K[2]^4+2 \text {a3} K[2]^3+c_1 K[2]^2+2 \text {a2} \log (K[2]) K[2]^2-2 \text {a1} K[2]-\text {a0}}}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{\sqrt {\text {a4} K[1]^4+2 \text {a3} K[1]^3-c_1 K[1]^2+2 \text {a2} \log (K[1]) K[1]^2-2 \text {a1} K[1]-\text {a0}}}dK[1]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{\sqrt {\text {a4} K[1]^4+2 \text {a3} K[1]^3+c_1 K[1]^2+2 \text {a2} \log (K[1]) K[1]^2-2 \text {a1} K[1]-\text {a0}}}dK[1]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {\text {a4} K[2]^4+2 \text {a3} K[2]^3-c_1 K[2]^2+2 \text {a2} \log (K[2]) K[2]^2-2 \text {a1} K[2]-\text {a0}}}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {\text {a4} K[2]^4+2 \text {a3} K[2]^3+c_1 K[2]^2+2 \text {a2} \log (K[2]) K[2]^2-2 \text {a1} K[2]-\text {a0}}}dK[2]\&\right ][x+c_2] \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a0 = symbols("a0")
a1 = symbols("a1")
a2 = symbols("a2")
a3 = symbols("a3")
a4 = symbols("a4")
y = Function("y")
ode = Eq(-a0 - a1*y(x) - a2*y(x)**2 - a3*y(x)**3 - a4*y(x)**4 + y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(-a0 - a1*y(x) - a2*y(x)**2 - a3*y(x)**3 - a4*y(x)**4 + y(x