54.1.184 problem 187

Internal problem ID [11498]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 187
Date solved : Tuesday, September 30, 2025 at 08:47:42 PM
CAS classification : [[_homogeneous, `class G`], _Riccati]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 59

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

\[ y = \frac {x^{n -1} \left (n -1+\tan \left (\frac {\sqrt {4 b a -n^{2}+2 n -1}\, \left (\ln \left (x \right )-c_1 \right )}{2}\right ) \sqrt {4 b a -n^{2}+2 n -1}\right )}{2 a} \]

✓ Mathematica. Time used: 0.302 (sec). Leaf size: 202

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

\begin{align*} y(x)&\to \frac {x^{n-1} \left (\left (-\sqrt {a} \sqrt {b} \sqrt {\frac {(n-1)^2}{a b}-4}+n-1\right ) x^{\sqrt {a} \sqrt {b} \sqrt {\frac {(n-1)^2}{a b}-4}}+c_1 \left (\sqrt {a} \sqrt {b} \sqrt {\frac {(n-1)^2}{a b}-4}+n-1\right )\right )}{2 a \left (x^{\sqrt {a} \sqrt {b} \sqrt {\frac {(n-1)^2}{a b}-4}}+c_1\right )}\\ y(x)&\to \frac {x^{n-1} \left (\sqrt {a} \sqrt {b} \sqrt {\frac {(n-1)^2}{a b}-4}+n-1\right )}{2 a} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*y(x)**2 - b*x**(2*n - 2) + x**n*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*x**(2*n - 2))/x**n cannot b