23.1.386 problem 371

Internal problem ID [4993]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 371
Date solved : Tuesday, September 30, 2025 at 09:13:15 AM
CAS classification : [[_homogeneous, `class D`], _rational, _Riccati]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 58

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

\[ y = -\tanh \left (\frac {c_1 \sqrt {4 a c -b^{2}}+2 \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right ) x \]

✓ Mathematica. Time used: 0.155 (sec). Leaf size: 53

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

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

✗ Sympy

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

Timed Out