23.4.41 problem 41
Internal
problem
ID
[6343]
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
:
41
Date
solved
:
Tuesday, September 30, 2025 at 02:51:58 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} b \sin \left (y\right )+a {y^{\prime }}^{2}+y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 211
ode:=b*sin(y(x))+a*diff(y(x),x)^2+diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
4 \int _{}^{y}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_1 \,a^{2}-4 a b \sin \left (\textit {\_a} \right )+2 \cos \left (\textit {\_a} \right ) b +{\mathrm e}^{-2 a \textit {\_a}} c_1 \right )}}d \textit {\_a} a^{2}+\int _{}^{y}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_1 \,a^{2}-4 a b \sin \left (\textit {\_a} \right )+2 \cos \left (\textit {\_a} \right ) b +{\mathrm e}^{-2 a \textit {\_a}} c_1 \right )}}d \textit {\_a} -c_2 -x &= 0 \\
-4 \int _{}^{y}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_1 \,a^{2}-4 a b \sin \left (\textit {\_a} \right )+2 \cos \left (\textit {\_a} \right ) b +{\mathrm e}^{-2 a \textit {\_a}} c_1 \right )}}d \textit {\_a} a^{2}-\int _{}^{y}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_1 \,a^{2}-4 a b \sin \left (\textit {\_a} \right )+2 \cos \left (\textit {\_a} \right ) b +{\mathrm e}^{-2 a \textit {\_a}} c_1 \right )}}d \textit {\_a} -c_2 -x &= 0 \\
\end{align*}
✓ Mathematica. Time used: 3.893 (sec). Leaf size: 444
ode=b*Sin[y[x]] + a*D[y[x],x]^2 + D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[1]} c_1 a^2-4 b \sin (K[1]) a+e^{-2 a K[1]} c_1+2 b \cos (K[1])}}dK[1]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[2]} c_1 a^2-4 b \sin (K[2]) a+e^{-2 a K[2]} c_1+2 b \cos (K[2])}}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[1]} (-c_1) a^2-4 b \sin (K[1]) a+e^{-2 a K[1]} (-c_1)+2 b \cos (K[1])}}dK[1]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[1]} c_1 a^2-4 b \sin (K[1]) a+e^{-2 a K[1]} c_1+2 b \cos (K[1])}}dK[1]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[2]} (-c_1) a^2-4 b \sin (K[2]) a+e^{-2 a K[2]} (-c_1)+2 b \cos (K[2])}}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[2]} c_1 a^2-4 b \sin (K[2]) a+e^{-2 a K[2]} c_1+2 b \cos (K[2])}}dK[2]\&\right ][x+c_2] \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(a*Derivative(y(x), x)**2 + b*sin(y(x)) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(-(b*sin(y(x)) + Derivative(y(x), (x, 2)))/a) + Derivative(