75.9.3 problem 222

Internal problem ID [16769]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8.3. The Lagrange and Clairaut equations. Exercises page 72
Problem number : 222
Date solved : Monday, March 31, 2025 at 03:16:41 PM
CAS classification : [_dAlembert]

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

✓ Maple. Time used: 0.082 (sec). Leaf size: 44

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

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

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 44

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

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

✗ Sympy

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

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