84.5.7 problem 3.2 (c)

Internal problem ID [22094]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 3. Classification of first-order differential equations. Solved problems. Page 11
Problem number : 3.2 (c)
Date solved : Thursday, October 02, 2025 at 08:23:57 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 45

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

\[ y = \operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}\frac {\sin \left (\frac {1}{\textit {\_a}}\right ) \textit {\_a}^{2}+1}{\textit {\_a} \left (\sin \left (\frac {1}{\textit {\_a}}\right ) \textit {\_a}^{2}-2 \,{\mathrm e}^{\textit {\_a}}+1\right )}d \textit {\_a} +\ln \left (x \right )+c_1 \right ) x \]

✓ Mathematica. Time used: 0.8 (sec). Leaf size: 60

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {\sin \left (\frac {1}{K[1]}\right ) K[1]^2+1}{K[1] \left (\sin \left (\frac {1}{K[1]}\right ) K[1]^2-2 e^{K[1]}+1\right )}dK[1]=-\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 21.486 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = C_{1} e^{- 2 \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {u_{1} e^{\frac {1}{u_{1}}}}{2 u_{1}^{2} e^{\frac {1}{u_{1}}} - u_{1}^{2} - \sin {\left (u_{1} \right )}}\, du_{1}} \]