55.1.5 problem 1.1.5
Internal
problem
ID
[13225]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
First-Order
differential
equations
Problem
number
:
1.1.5
Date
solved
:
Wednesday, October 01, 2025 at 03:37:17 AM
CAS
classification
:
[_Bernoulli]
\begin{align*} g \left (x \right ) y^{\prime }&=f_{1} \left (x \right ) y+f_{n} \left (x \right ) y^{n} \end{align*}
✓ Maple. Time used: 0.049 (sec). Leaf size: 74
ode:=g(x)*diff(y(x),x) = f__1(x)*y(x)+f__n(x)*y(x)^n;
dsolve(ode,y(x), singsol=all);
\[
y = {\mathrm e}^{\int \frac {f_{1} \left (x \right )}{g \left (x \right )}d x} \left (-n \int \frac {f_{n} \left (x \right ) {\mathrm e}^{\left (n -1\right ) \int \frac {f_{1} \left (x \right )}{g \left (x \right )}d x}}{g \left (x \right )}d x +c_1 +\int \frac {f_{n} \left (x \right ) {\mathrm e}^{\left (n -1\right ) \int \frac {f_{1} \left (x \right )}{g \left (x \right )}d x}}{g \left (x \right )}d x \right )^{-\frac {1}{n -1}}
\]
✓ Mathematica. Time used: 9.106 (sec). Leaf size: 163
ode=g[x]*D[y[x],x]==f1[x]*y[x]+fn[x]*y[x]^n;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \left (\exp \left (-\left ((n-1) \int _1^x\frac {\text {f1}(K[1])}{g(K[1])}dK[1]\right )\right ) \left (-(n-1) \int _1^x\frac {\exp \left ((n-1) \int _1^{K[2]}\frac {\text {f1}(K[1])}{g(K[1])}dK[1]\right ) \text {fn}(K[2])}{g(K[2])}dK[2]+c_1\right )\right ){}^{\frac {1}{1-n}}\\ y(x)&\to \left ((n-1) \left (-\exp \left (-\left ((n-1) \int _1^x\frac {\text {f1}(K[1])}{g(K[1])}dK[1]\right )\right )\right ) \int _1^x\frac {\exp \left ((n-1) \int _1^{K[2]}\frac {\text {f1}(K[1])}{g(K[1])}dK[1]\right ) \text {fn}(K[2])}{g(K[2])}dK[2]\right ){}^{\frac {1}{1-n}} \end{align*}
✓ Sympy. Time used: 49.963 (sec). Leaf size: 87
from sympy import *
x = symbols("x")
n = symbols("n")
y = Function("y")
f1 = Function("f1")
fn = Function("fn")
g = Function("g")
ode = Eq(-f1(x)*y(x) - fn(x)*y(x)**n + g(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (\left (C_{1} - n \int \frac {\operatorname {fn}{\left (x \right )} \left (e^{n \int \frac {f_{1}{\left (x \right )}}{g{\left (x \right )}}\, dx}\right ) e^{- \int \frac {f_{1}{\left (x \right )}}{g{\left (x \right )}}\, dx}}{g{\left (x \right )}}\, dx + \int \frac {\operatorname {fn}{\left (x \right )} \left (e^{n \int \frac {f_{1}{\left (x \right )}}{g{\left (x \right )}}\, dx}\right ) e^{- \int \frac {f_{1}{\left (x \right )}}{g{\left (x \right )}}\, dx}}{g{\left (x \right )}}\, dx\right ) e^{- \left (n - 1\right ) \int \frac {f_{1}{\left (x \right )}}{g{\left (x \right )}}\, dx}\right )^{- \frac {1}{n - 1}}
\]