29.19.2 problem 515

Internal problem ID [5111]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 19
Problem number : 515
Date solved : Sunday, March 30, 2025 at 06:41:03 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x y y^{\prime }+x^{2} \operatorname {arccot}\left (\frac {y}{x}\right )-y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 21

ode:=x*y(x)*diff(y(x),x)+x^2*arccot(y(x)/x)-y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}}{\operatorname {arccot}\left (\textit {\_a} \right )}d \textit {\_a} +\ln \left (x \right )+c_1 \right ) x \]

✓ Mathematica. Time used: 0.594 (sec). Leaf size: 31

ode=x y[x] D[y[x],x]+x^2 ArcCot[y[x]/x]-y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 41.875 (sec). Leaf size: 66

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

\[ y{\left (x \right )} = C_{1} e^{- 2 \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1}^{3} \left (2 \operatorname {atan}{\left (\frac {1}{u_{1}} \right )} - \pi \right )}\, du_{1} + \pi \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} \left (2 \operatorname {atan}{\left (\frac {1}{u_{1}} \right )} - \pi \right )}\, du_{1} - 2 \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\operatorname {atan}{\left (\frac {1}{u_{1}} \right )}}{u_{1} \left (2 \operatorname {atan}{\left (\frac {1}{u_{1}} \right )} - \pi \right )}\, du_{1}} \]