55.2.60 problem 60

Internal problem ID [13286]
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 : 60
Date solved : Wednesday, October 01, 2025 at 05:47:10 AM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 6414

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

\[ \text {Expression too large to display} \]

✓ Mathematica. Time used: 5.843 (sec). Leaf size: 245

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

\begin{align*} y(x)&\to \frac {c \left (-\exp \left (\int _1^x\frac {-b+\mu -a K[1]+2 \lambda K[1]}{c+K[1] (b+a K[1])}dK[1]\right )\right )+\lambda x \int _1^x\exp \left (\int _1^{K[2]}\frac {-b+\mu -a K[1]+2 \lambda K[1]}{c+K[1] (b+a K[1])}dK[1]\right )dK[2]-x \left (b \exp \left (\int _1^x\frac {-b+\mu -a K[1]+2 \lambda K[1]}{c+K[1] (b+a K[1])}dK[1]\right )+a x \exp \left (\int _1^x\frac {-b+\mu -a K[1]+2 \lambda K[1]}{c+K[1] (b+a K[1])}dK[1]\right )-c_1 \lambda \right )}{\int _1^x\exp \left (\int _1^{K[2]}\frac {-b+\mu -a K[1]+2 \lambda K[1]}{c+K[1] (b+a K[1])}dK[1]\right )dK[2]+c_1}\\ y(x)&\to \lambda x \end{align*}

✗ Sympy

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

Timed Out