55.1.3 problem 1.1.3

Internal problem ID [13223]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, First-Order differential equations
Problem number : 1.1.3
Date solved : Wednesday, October 01, 2025 at 03:37:14 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 20

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

\[ \int f \left (x \right )d x -\int _{}^{y}\frac {1}{g \left (\textit {\_a} \right )}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 0.135 (sec). Leaf size: 42

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{g(K[1])}dK[1]\&\right ]\left [\int _1^xf(K[2])dK[2]+c_1\right ]\\ y(x)&\to g^{(-1)}(0) \end{align*}

✓ Sympy. Time used: 0.155 (sec). Leaf size: 14

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

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