80.3.30 problem 32

Internal problem ID [21194]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 3. First order nonlinear differential equations. Excercise 3.7 at page 67
Problem number : 32
Date solved : Thursday, October 02, 2025 at 07:26:08 PM
CAS classification : [_Bernoulli]

\begin{align*} x +y^{2}+B \left (x \right ) y y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 94

ode:=x+y(x)^2+B(x)*y(x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {{\mathrm e}^{2 \int \frac {1}{B \left (x \right )}d x} \left (-2 \int \frac {{\mathrm e}^{2 \int \frac {1}{B \left (x \right )}d x} x}{B \left (x \right )}d x +c_1 \right )}\, {\mathrm e}^{-2 \int \frac {1}{B \left (x \right )}d x} \\ y &= -\sqrt {{\mathrm e}^{2 \int \frac {1}{B \left (x \right )}d x} \left (-2 \int \frac {{\mathrm e}^{2 \int \frac {1}{B \left (x \right )}d x} x}{B \left (x \right )}d x +c_1 \right )}\, {\mathrm e}^{-2 \int \frac {1}{B \left (x \right )}d x} \\ \end{align*}

✓ Mathematica. Time used: 0.195 (sec). Leaf size: 132

ode=(x+y[x]^2)+(B[x]*y[x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 10.819 (sec). Leaf size: 73

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

\[ \left [ y{\left (x \right )} = - \sqrt {\left (C_{1} - 2 \int \frac {x e^{2 \int \frac {1}{B{\left (x \right )}}\, dx}}{B{\left (x \right )}}\, dx\right ) e^{- 2 \int \frac {1}{B{\left (x \right )}}\, dx}}, \ y{\left (x \right )} = \sqrt {\left (C_{1} - 2 \int \frac {x e^{2 \int \frac {1}{B{\left (x \right )}}\, dx}}{B{\left (x \right )}}\, dx\right ) e^{- 2 \int \frac {1}{B{\left (x \right )}}\, dx}}\right ] \]