75.9.1 problem 220
Internal
problem
ID
[16767]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
8.3.
The
Lagrange
and
Clairaut
equations.
Exercises
page
72
Problem
number
:
220
Date
solved
:
Monday, March 31, 2025 at 03:16:35 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} y&=2 x y^{\prime }+\ln \left (y^{\prime }\right ) \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 69
ode:=y(x) = 2*x*diff(y(x),x)+ln(diff(y(x),x));
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: 0.07 (sec). Leaf size: 258
ode=y[x]==2*x*D[y[x],x]+Log[D[y[x],x]];
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: 1.073 (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)
\[
C_{1} - y{\left (x \right )} - \log {\left (W\left (2 x e^{y{\left (x \right )}}\right ) + 2 \right )} + W\left (2 x e^{y{\left (x \right )}}\right ) = 0
\]