60.1.263 problem 268
Internal
problem
ID
[10277]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
268
Date
solved
:
Sunday, March 30, 2025 at 03:42:27 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} f \left (x \right ) y y^{\prime }+g \left (x \right ) y^{2}+h \left (x \right )&=0 \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 114
ode:=f(x)*y(x)*diff(y(x),x)+g(x)*y(x)^2+h(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \sqrt {{\mathrm e}^{\int \frac {2 g \left (x \right )}{f \left (x \right )}d x} \left (-2 \int \frac {{\mathrm e}^{\int \frac {2 g \left (x \right )}{f \left (x \right )}d x} h \left (x \right )}{f \left (x \right )}d x +c_1 \right )}\, {\mathrm e}^{-2 \int \frac {g \left (x \right )}{f \left (x \right )}d x} \\
y &= -\sqrt {{\mathrm e}^{\int \frac {2 g \left (x \right )}{f \left (x \right )}d x} \left (-2 \int \frac {{\mathrm e}^{\int \frac {2 g \left (x \right )}{f \left (x \right )}d x} h \left (x \right )}{f \left (x \right )}d x +c_1 \right )}\, {\mathrm e}^{-2 \int \frac {g \left (x \right )}{f \left (x \right )}d x} \\
\end{align*}
✓ Mathematica. Time used: 0.361 (sec). Leaf size: 146
ode=f[x]*y[x]*D[y[x],x]+g[x]*y[x]^2+h[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\exp \left (\int _1^x-\frac {g(K[1])}{f(K[1])}dK[1]\right ) \sqrt {2 \int _1^x-\frac {\exp \left (-2 \int _1^{K[2]}-\frac {g(K[1])}{f(K[1])}dK[1]\right ) h(K[2])}{f(K[2])}dK[2]+c_1} \\
y(x)\to \exp \left (\int _1^x-\frac {g(K[1])}{f(K[1])}dK[1]\right ) \sqrt {2 \int _1^x-\frac {\exp \left (-2 \int _1^{K[2]}-\frac {g(K[1])}{f(K[1])}dK[1]\right ) h(K[2])}{f(K[2])}dK[2]+c_1} \\
\end{align*}
✓ Sympy. Time used: 41.767 (sec). Leaf size: 83
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(f(x)*y(x)*Derivative(y(x), x) + g(x)*y(x)**2 + h(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \sqrt {\left (C_{1} - 2 \int \frac {h{\left (x \right )} e^{2 \int \frac {g{\left (x \right )}}{f{\left (x \right )}}\, dx}}{f{\left (x \right )}}\, dx\right ) e^{- 2 \int \frac {g{\left (x \right )}}{f{\left (x \right )}}\, dx}}, \ y{\left (x \right )} = \sqrt {\left (C_{1} - 2 \int \frac {h{\left (x \right )} e^{2 \int \frac {g{\left (x \right )}}{f{\left (x \right )}}\, dx}}{f{\left (x \right )}}\, dx\right ) e^{- 2 \int \frac {g{\left (x \right )}}{f{\left (x \right )}}\, dx}}\right ]
\]