55.2.61 problem 61

Internal problem ID [13287]
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 : 61
Date solved : Wednesday, October 01, 2025 at 05:48:43 AM
CAS classification : [_rational, _Riccati]

\begin{align*} \left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime }&=y^{2}+\left (a_{1} x +b_{1} \right ) y-\lambda \left (\lambda +a_{1} -a_{2} \right ) x^{2}+\lambda \left (b_{2} -b_{1} \right ) x +\lambda c_{2} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 5779

ode:=(a__2*x^2+b__2*x+c__2)*diff(y(x),x) = y(x)^2+(a__1*x+b__1)*y(x)-lambda*(lambda+a__1-a__2)*x^2+lambda*(b__2-b__1)*x+lambda*c__2; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 7.145 (sec). Leaf size: 250

ode=(a2*x^2+b2*x+c2)*D[y[x],x]==y[x]^2+(a1*x+b1)*y[x]-\[Lambda]*(\[Lambda]+a1-a2)*x^2+\[Lambda]*(b2-b1)*x+\[Lambda]*c2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\text {c2} \left (-\exp \left (\int _1^x\frac {\text {b1}-\text {b2}+(\text {a1}-2 \text {a2}+2 \lambda ) K[1]}{\text {c2}+K[1] (\text {b2}+\text {a2} K[1])}dK[1]\right )\right )+\lambda x \int _1^x\exp \left (\int _1^{K[2]}\frac {\text {b1}-\text {b2}+(\text {a1}-2 \text {a2}+2 \lambda ) K[1]}{\text {c2}+K[1] (\text {b2}+\text {a2} K[1])}dK[1]\right )dK[2]-x \left (\text {b2} \exp \left (\int _1^x\frac {\text {b1}-\text {b2}+(\text {a1}-2 \text {a2}+2 \lambda ) K[1]}{\text {c2}+K[1] (\text {b2}+\text {a2} K[1])}dK[1]\right )+\text {a2} x \exp \left (\int _1^x\frac {\text {b1}-\text {b2}+(\text {a1}-2 \text {a2}+2 \lambda ) K[1]}{\text {c2}+K[1] (\text {b2}+\text {a2} K[1])}dK[1]\right )-c_1 \lambda \right )}{\int _1^x\exp \left (\int _1^{K[2]}\frac {\text {b1}-\text {b2}+(\text {a1}-2 \text {a2}+2 \lambda ) K[1]}{\text {c2}+K[1] (\text {b2}+\text {a2} K[1])}dK[1]\right )dK[2]+c_1}\\ y(x)&\to \lambda x \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a__1 = symbols("a__1") 
a__2 = symbols("a__2") 
b__1 = symbols("b__1") 
b__2 = symbols("b__2") 
c__2 = symbols("c__2") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(-c__2*lambda_ + lambda_*x**2*(a__1 - a__2 + lambda_) - lambda_*x*(-b__1 + b__2) - (a__1*x + b__1)*y(x) + (a__2*x**2 + b__2*x + c__2)*Derivative(y(x), x) - y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out