54.2.392 problem 971

Internal problem ID [12266]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 971
Date solved : Wednesday, October 01, 2025 at 01:26:25 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]

\begin{align*} y^{\prime }&=\frac {\left (x y+1\right )^{3}}{x^{5}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 70

ode:=diff(y(x),x) = (x*y(x)+1)^3/x^5; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-2+x^{3} \left (\tan \left (\operatorname {RootOf}\left (18 x^{3} \left (-\frac {1}{x^{6}}\right )^{{2}/{3}}+6 \textit {\_Z} \sqrt {3}-3 \ln \left (3\right )+\ln \left (\left (\sqrt {3}\, \sin \left (\textit {\_Z} \right )+3 \cos \left (\textit {\_Z} \right )\right )^{6}\right )-18 c_1 \right )\right ) \sqrt {3}+1\right ) \left (-\frac {1}{x^{6}}\right )^{{1}/{3}}}{2 x} \]

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 61

ode=D[y[x],x] == (1 + x*y[x])^3/x^5; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {\frac {3 y(x)}{x^2}+\frac {3}{x^3}}{3 \sqrt [3]{-\frac {1}{x^6}}}}\frac {1}{K[1]^3+1}dK[1]=-\left (-\frac {1}{x^6}\right )^{2/3} x^3+c_1,y(x)\right ] \]

✓ Sympy. Time used: 3.004 (sec). Leaf size: 63

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x*y(x) + 1)**3/x**5,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - \frac {\log {\left (y{\left (x \right )} - 1 + \frac {1}{x} \right )}}{3} + \frac {\log {\left (\left (y{\left (x \right )} + \frac {1}{x}\right )^{2} + y{\left (x \right )} + 1 + \frac {1}{x} \right )}}{6} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \left (2 y{\left (x \right )} + 1 + \frac {2}{x}\right )}{3} \right )}}{3} - \frac {1}{x} = 0 \]