29.15.25 problem 433
Internal
problem
ID
[5031]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
15
Problem
number
:
433
Date
solved
:
Sunday, March 30, 2025 at 06:31:05 AM
CAS
classification
:
[_separable]
\begin{align*} \left (1+y\right ) y^{\prime }&=x^{2} \left (1-y\right ) \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 20
ode:=(1+y(x))*diff(y(x),x) = x^2*(1-y(x));
dsolve(ode,y(x), singsol=all);
\[
y = 2 \operatorname {LambertW}\left (\frac {c_1 \,{\mathrm e}^{-\frac {x^{3}}{6}-\frac {1}{2}}}{2}\right )+1
\]
✓ Mathematica. Time used: 22.285 (sec). Leaf size: 66
ode=(1+y[x])D[y[x],x]==x^2(1-y[x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to 1+2 W\left (-\frac {1}{2} \sqrt {e^{-\frac {x^3}{3}-1+c_1}}\right ) \\
y(x)\to 1+2 W\left (\frac {1}{2} \sqrt {e^{-\frac {x^3}{3}-1+c_1}}\right ) \\
y(x)\to 1 \\
\end{align*}
✓ Sympy. Time used: 10.255 (sec). Leaf size: 180
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2*(1 - y(x)) + (y(x) + 1)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = 2 W\left (- \frac {\sqrt [6]{C_{1} e^{- x^{3}}}}{2 e^{\frac {1}{2}}}\right ) + 1, \ y{\left (x \right )} = 2 W\left (\frac {\sqrt [6]{C_{1} e^{- x^{3}}}}{2 e^{\frac {1}{2}}}\right ) + 1, \ y{\left (x \right )} = 2 W\left (\frac {\sqrt [6]{C_{1} e^{- x^{3}}} \left (-1 - \sqrt {3} i\right )}{4 e^{\frac {1}{2}}}\right ) + 1, \ y{\left (x \right )} = 2 W\left (\frac {\sqrt [6]{C_{1} e^{- x^{3}}} \left (-1 + \sqrt {3} i\right )}{4 e^{\frac {1}{2}}}\right ) + 1, \ y{\left (x \right )} = 2 W\left (\frac {\sqrt [6]{C_{1} e^{- x^{3}}} \left (1 - \sqrt {3} i\right )}{4 e^{\frac {1}{2}}}\right ) + 1, \ y{\left (x \right )} = 2 W\left (\frac {\sqrt [6]{C_{1} e^{- x^{3}}} \left (1 + \sqrt {3} i\right )}{4 e^{\frac {1}{2}}}\right ) + 1\right ]
\]