61.2.77 problem 77

Internal problem ID [12004]
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 : 77
Date solved : Sunday, March 30, 2025 at 10:11:52 PM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 164

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

\[ y = \frac {-\int \frac {x^{k} {\mathrm e}^{-\int \frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x}}{a \,x^{n}+b \,x^{m}+c}d x \alpha \lambda -c_1 \lambda -{\mathrm e}^{-\int \frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x}}{c_1 +\alpha \int \frac {x^{k} {\mathrm e}^{-\int \frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x}}{a \,x^{n}+b \,x^{m}+c}d x} \]

✓ Mathematica. Time used: 4.922 (sec). Leaf size: 389

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

\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) \left (-\alpha \lambda K[2]^k+\alpha y(x) K[2]^k+\beta K[2]^s\right )}{(k-s) \alpha \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) K[2]^k}{(k-s) \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +K[3])}-\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) \left (-\alpha \lambda K[2]^k+\alpha K[3] K[2]^k+\beta K[2]^s\right )}{(k-s) \alpha \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +K[3])^2}\right )dK[2]-\frac {\exp \left (-\int _1^x-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right )}{(k-s) \alpha \beta (\lambda +K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out