54.1.225 problem 230

Internal problem ID [11539]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 230
Date solved : Tuesday, September 30, 2025 at 08:58:33 PM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 98

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

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

✓ Mathematica. Time used: 0.214 (sec). Leaf size: 98

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

\begin{align*} y(x)&\to -e^{-\frac {b x}{a}} \sqrt {2 \int _1^x-\frac {e^{\frac {2 b K[1]}{a}} f(K[1])}{a}dK[1]+c_1}\\ y(x)&\to e^{-\frac {b x}{a}} \sqrt {2 \int _1^x-\frac {e^{\frac {2 b K[1]}{a}} f(K[1])}{a}dK[1]+c_1} \end{align*}

✓ Sympy. Time used: 2.236 (sec). Leaf size: 66

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

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