29.37.17 problem 1137

Internal problem ID [5675]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 37
Problem number : 1137
Date solved : Sunday, March 30, 2025 at 09:59:49 AM
CAS classification : [_dAlembert]

\begin{align*} {y^{\prime }}^{2} \left (x +\sin \left (y^{\prime }\right )\right )&=y \end{align*}

✓ Maple. Time used: 0.306 (sec). Leaf size: 56

ode:=diff(y(x),x)^2*(x+sin(diff(y(x),x))) = y(x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ \left [x \left (\textit {\_T} \right ) &= \frac {\left (-\textit {\_T}^{2}+\textit {\_T} \right ) \sin \left (\textit {\_T} \right )+c_1 -\cos \left (\textit {\_T} \right )}{\left (\textit {\_T} -1\right )^{2}}, y \left (\textit {\_T} \right ) = -\frac {\left (\left (\textit {\_T} -1\right ) \sin \left (\textit {\_T} \right )-c_1 +\cos \left (\textit {\_T} \right )\right ) \textit {\_T}^{2}}{\left (\textit {\_T} -1\right )^{2}}\right ] \\ \end{align*}

✓ Mathematica. Time used: 0.144 (sec). Leaf size: 81

ode=(D[y[x],x])^2*(x+Sin[D[y[x],x]])==y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\left \{x=\frac {\int \frac {(K[1]-1)^2 \left (2 K[1] \sin (K[1])+K[1]^2 \cos (K[1])\right )}{K[1]-K[1]^2} \, dK[1]}{(K[1]-1)^2}+\frac {c_1}{(K[1]-1)^2},y(x)=x K[1]^2+K[1]^2 \sin (K[1])\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

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

NotImplementedError : multiple generators [_X0, sin(_X0)] 
No algorithms are implemented to solve equation _X0**2*x + _X0**2*sin(_X0) - y(x)