82.15.1 problem Ex. 1
Internal
problem
ID
[18740]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
III.
Equations
of
the
first
order
but
not
of
the
first
degree.
Problems
at
page
35
Problem
number
:
Ex.
1
Date
solved
:
Monday, March 31, 2025 at 06:07:18 PM
CAS
classification
:
[_quadrature]
\begin{align*} y&=2 y^{\prime }+3 {y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.033 (sec). Leaf size: 106
ode:=y(x) = 2*diff(y(x),x)+3*diff(y(x),x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\operatorname {LambertW}\left (-\sqrt {3}\, {\mathrm e}^{-1+\frac {x}{2}-\frac {c_1}{2}}\right ) \left (\operatorname {LambertW}\left (-\sqrt {3}\, {\mathrm e}^{-1+\frac {x}{2}-\frac {c_1}{2}}\right )+2\right )}{3} \\
y &= \frac {{\mathrm e}^{2 \operatorname {RootOf}\left (-\textit {\_Z} -x +2 \,{\mathrm e}^{\textit {\_Z}}-2+c_1 -\ln \left (3\right )+\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}-2\right )^{2}\right )\right )}}{3}-\frac {2 \,{\mathrm e}^{\operatorname {RootOf}\left (-\textit {\_Z} -x +2 \,{\mathrm e}^{\textit {\_Z}}-2+c_1 -\ln \left (3\right )+\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}-2\right )^{2}\right )\right )}}{3} \\
\end{align*}
✓ Mathematica. Time used: 18.973 (sec). Leaf size: 86
ode=y[x]==2*D[y[x],x]+3*D[y[x],x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{3} W\left (-e^{\frac {1}{2} (x-2-3 c_1)}\right ) \left (2+W\left (-e^{\frac {1}{2} (x-2-3 c_1)}\right )\right ) \\
y(x)\to \frac {1}{3} W\left (e^{\frac {1}{2} (x-2+3 c_1)}\right ) \left (2+W\left (e^{\frac {1}{2} (x-2+3 c_1)}\right )\right ) \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 0.703 (sec). Leaf size: 58
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(y(x) - 3*Derivative(y(x), x)**2 - 2*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ x + 2 \sqrt {3 y{\left (x \right )} + 1} - 2 \log {\left (\sqrt {3 y{\left (x \right )} + 1} + 1 \right )} = C_{1}, \ x - 2 \sqrt {3 y{\left (x \right )} + 1} - 2 \log {\left (\sqrt {3 y{\left (x \right )} + 1} - 1 \right )} = C_{1}\right ]
\]