89.2.17 problem 17

Internal problem ID [24282]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 27
Problem number : 17
Date solved : Thursday, October 02, 2025 at 10:06:55 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x -y \arctan \left (\frac {y}{x}\right )+x \arctan \left (\frac {y}{x}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.269 (sec). Leaf size: 59

ode:=x-y(x)*arctan(y(x)/x)+x*arctan(y(x)/x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \tan \left (\operatorname {RootOf}\left (\cos \left (\textit {\_Z} \right )^{2} \ln \left (\frac {\sec \left (\textit {\_Z} \right )^{2}}{x^{2}}\right )^{2}-4 \cos \left (\textit {\_Z} \right )^{2} \ln \left (\frac {\sec \left (\textit {\_Z} \right )^{2}}{x^{2}}\right ) c_1 +4 \cos \left (\textit {\_Z} \right )^{2} c_1^{2}-4 \sin \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}\right )\right ) x \]

✓ Mathematica. Time used: 0.12 (sec). Leaf size: 40

ode=(x-y[x]*ArcTan[y[x]/x])+(x*ArcTan[y[x]/x] )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {y(x) \arctan \left (\frac {y(x)}{x}\right )}{x}-\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )=-\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.327 (sec). Leaf size: 31

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

\[ \log {\left (x \right )} = C_{1} + \log {\left (\sqrt {1 + \frac {y^{2}{\left (x \right )}}{x^{2}}} \right )} - \frac {y{\left (x \right )} \operatorname {atan}{\left (\frac {y{\left (x \right )}}{x} \right )}}{x} \]