60.1.96 problem 98

Internal problem ID [10110]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 98
Date solved : Sunday, March 30, 2025 at 03:16:25 PM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }+a y^{2}-b y+c \,x^{2 b}&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 34

ode:=x*diff(y(x),x)+a*y(x)^2-b*y(x)+c*x^(2*b) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\tan \left (\frac {\sqrt {a}\, \sqrt {c}\, x^{b}+c_1 b}{b}\right ) \sqrt {c}\, x^{b}}{\sqrt {a}} \]

✓ Mathematica. Time used: 0.346 (sec). Leaf size: 211

ode=x*D[y[x],x] + a*y[x]^2 - b*y[x] + c*x^(2*b)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {\sqrt {-c} x^b \left (-\cos \left (\frac {\sqrt {-a} \sqrt {-c} x^b}{b}\right )+c_1 \sin \left (\frac {\sqrt {-a} \sqrt {-c} x^b}{b}\right )\right )}{\sqrt {-a} \left (\sin \left (\frac {\sqrt {-a} \sqrt {-c} x^b}{b}\right )+c_1 \cos \left (\frac {\sqrt {-a} \sqrt {-c} x^b}{b}\right )\right )} \\ y(x)\to \frac {\sqrt {-c} x^b \tan \left (\frac {\sqrt {-a} \sqrt {-c} x^b}{b}\right )}{\sqrt {-a}} \\ y(x)\to \frac {\sqrt {-c} x^b \tan \left (\frac {\sqrt {-a} \sqrt {-c} x^b}{b}\right )}{\sqrt {-a}} \\ \end{align*}

✗ Sympy

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

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