61.2.78 problem 78

Internal problem ID [12005]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number : 78
Date solved : Sunday, March 30, 2025 at 10:14:18 PM
CAS classification : [[_homogeneous, `class D`], _rational, _Riccati]

\begin{align*} \left (a \,x^{n}+b \,x^{m}+c \right ) \left (x y^{\prime }-y\right )+s \,x^{k} \left (y^{2}-\lambda \,x^{2}\right )&=0 \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 37

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

\[ y = \tanh \left (s \sqrt {\lambda }\, \left (\int \frac {x^{k}}{a \,x^{n}+b \,x^{m}+c}d x +c_1 \right )\right ) x \sqrt {\lambda } \]

✓ Mathematica. Time used: 0.75 (sec). Leaf size: 59

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {1}{K[1]^2-\lambda }dK[1]=\int _1^x-\frac {s K[2]^k}{b K[2]^m+a K[2]^n+c}dK[2]+c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out