62.14.1 problem Ex 1
Internal
problem
ID
[12811]
Book
:
An
elementary
treatise
on
differential
equations
by
Abraham
Cohen.
DC
heath
publishers.
1906
Section
:
Chapter
IV,
differential
equations
of
the
first
order
and
higher
degree
than
the
first.
Article
25.
Equations
solvable
for
\(y\).
Page
52
Problem
number
:
Ex
1
Date
solved
:
Monday, March 31, 2025 at 07:10:30 AM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} 2 y^{\prime } x -y+\ln \left (y^{\prime }\right )&=0 \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 69
ode:=2*x*diff(y(x),x)-y(x)+ln(diff(y(x),x)) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -1+\sqrt {4 c_1 x +1}-\ln \left (2\right )+\ln \left (\frac {-1+\sqrt {4 c_1 x +1}}{x}\right ) \\
y &= -1-\sqrt {4 c_1 x +1}-\ln \left (2\right )+\ln \left (\frac {-1-\sqrt {4 c_1 x +1}}{x}\right ) \\
\end{align*}
✓ Mathematica. Time used: 0.143 (sec). Leaf size: 258
ode=2*D[y[x],x]*x-y[x]+Log[D[y[x],x]]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^x\frac {W\left (2 e^{y(x)} K[1]\right )}{K[1] \left (W\left (2 e^{y(x)} K[1]\right )+2\right )}dK[1]+\int _1^{y(x)}-\frac {W\left (2 e^{K[2]} x\right ) \int _1^x\left (\frac {W\left (2 e^{K[2]} K[1]\right )}{K[1] \left (W\left (2 e^{K[2]} K[1]\right )+1\right ) \left (W\left (2 e^{K[2]} K[1]\right )+2\right )}-\frac {W\left (2 e^{K[2]} K[1]\right )^2}{K[1] \left (W\left (2 e^{K[2]} K[1]\right )+1\right ) \left (W\left (2 e^{K[2]} K[1]\right )+2\right )^2}\right )dK[1]+2 \int _1^x\left (\frac {W\left (2 e^{K[2]} K[1]\right )}{K[1] \left (W\left (2 e^{K[2]} K[1]\right )+1\right ) \left (W\left (2 e^{K[2]} K[1]\right )+2\right )}-\frac {W\left (2 e^{K[2]} K[1]\right )^2}{K[1] \left (W\left (2 e^{K[2]} K[1]\right )+1\right ) \left (W\left (2 e^{K[2]} K[1]\right )+2\right )^2}\right )dK[1]+2}{W\left (2 e^{K[2]} x\right )+2}dK[2]=c_1,y(x)\right ]
\]
✓ Sympy. Time used: 1.146 (sec). Leaf size: 29
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*Derivative(y(x), x) - y(x) + log(Derivative(y(x), x)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
C_{1} - y{\left (x \right )} - \log {\left (W\left (2 x e^{y{\left (x \right )}}\right ) + 2 \right )} + W\left (2 x e^{y{\left (x \right )}}\right ) = 0
\]