60.1.34 problem 34

Internal problem ID [10048]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 34
Date solved : Sunday, March 30, 2025 at 02:55:34 PM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 28

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

\[ y = \frac {{\mathrm e}^{-\int g \left (x \right )d x}}{\int {\mathrm e}^{-\int g \left (x \right )d x} f \left (x \right )d x +c_1} \]

✓ Mathematica. Time used: 0.174 (sec). Leaf size: 59

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

\begin{align*} y(x)\to \frac {\exp \left (\int _1^x-g(K[1])dK[1]\right )}{-\int _1^x-\exp \left (\int _1^{K[2]}-g(K[1])dK[1]\right ) f(K[2])dK[2]+c_1} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.895 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = \frac {e^{- \int g{\left (x \right )}\, dx}}{C_{1} + \int f{\left (x \right )} e^{- \int g{\left (x \right )}\, dx}\, dx} \]