60.2.105 problem 681

Internal problem ID [10679]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 681
Date solved : Sunday, March 30, 2025 at 06:20:27 PM
CAS classification : [[_homogeneous, `class D`], _Riccati]

\begin{align*} y^{\prime }&=\frac {y+x^{3} b \ln \left (\frac {1}{x}\right )+x^{4} b +b \,x^{3}+x a y^{2} \ln \left (\frac {1}{x}\right )+x^{2} a y^{2}+a x y^{2}}{x} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 45

ode:=diff(y(x),x) = (y(x)+x^3*b*ln(1/x)+x^4*b+b*x^3+x*a*y(x)^2*ln(1/x)+x^2*a*y(x)^2+a*x*y(x)^2)/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\tan \left (\frac {\left (6 x^{2} \ln \left (\frac {1}{x}\right )+4 x^{3}+9 x^{2}+12 c_1 \right ) \sqrt {b a}}{12}\right ) x \sqrt {b a}}{a} \]

✓ Mathematica. Time used: 0.16 (sec). Leaf size: 52

ode=D[y[x],x] == (b*x^3 + b*x^4 + b*x^3*Log[x^(-1)] + y[x] + a*x*y[x]^2 + a*x^2*y[x]^2 + a*x*Log[x^(-1)]*y[x]^2)/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {1}{a K[1]^2+b}dK[1]=\frac {x^3}{3}+\frac {3 x^2}{4}-\frac {1}{2} x^2 \log (x)+c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (a*x**2*y(x)**2 + a*x*y(x)**2*log(1/x) + a*x*y(x)**2 + b*x**4 + b*x**3*log(1/x) + b*x**3 + y(x))/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*x*y(x)**2 - a*y(x)**2*log(1/x) - a*y(x)**2 - b*x**3 - b*x**2*log(1/x) - b*x**2 + Derivative(y(x), x) - y(x)/x cannot be solved by the factorable group method