23.1.231 problem 227

Internal problem ID [4838]
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 : 227
Date solved : Tuesday, September 30, 2025 at 08:43:49 AM
CAS classification : [_rational, _Bernoulli]

\begin{align*} \left (1+x \right ) y^{\prime }&=a y+b x y^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 31

ode:=(1+x)*diff(y(x),x) = a*y(x)+b*x*y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {a \left (a +1\right )}{a c_1 \left (a +1\right ) \left (1+x \right )^{-a}-b x a +b} \]

✓ Mathematica. Time used: 0.71 (sec). Leaf size: 44

ode=(1+x)*D[y[x],x]==a*y[x]+b*x*y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {a (a+1) (x+1)^a}{b (x+1)^a (a x-1)-a (a+1) c_1}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.483 (sec). Leaf size: 126

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-a*y(x) - b*x*y(x)**2 + (x + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \begin {cases} \frac {1}{C_{1} x + C_{1} - b x \log {\left (x + 1 \right )} - b \log {\left (x + 1 \right )} - b} & \text {for}\: a = -1 \\\frac {1}{C_{1} - b x + b \log {\left (x + 1 \right )}} & \text {for}\: a = 0 \\\frac {a^{2} e^{a \log {\left (x + 1 \right )}}}{C_{1} a^{2} + C_{1} a - a b x e^{a \log {\left (x + 1 \right )}} + b e^{a \log {\left (x + 1 \right )}}} + \frac {a e^{a \log {\left (x + 1 \right )}}}{C_{1} a^{2} + C_{1} a - a b x e^{a \log {\left (x + 1 \right )}} + b e^{a \log {\left (x + 1 \right )}}} & \text {otherwise} \end {cases} \]