23.2.352 problem 403
Internal
problem
ID
[5707]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
2.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
SECOND
OR
HIGHER
DEGREE,
page
278
Problem
number
:
403
Date
solved
:
Tuesday, September 30, 2025 at 02:01:29 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} \ln \left (y^{\prime }\right )+x y^{\prime }+a +b y&=0 \end{align*}
✓ Maple. Time used: 0.176 (sec). Leaf size: 54
ode:=ln(diff(y(x),x))+x*diff(y(x),x)+a+b*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
-{\left (\frac {\operatorname {LambertW}\left (x \,{\mathrm e}^{-b y-a}\right )}{x}\right )}^{-\frac {1}{b +1}} c_1 +x -\frac {x}{b \operatorname {LambertW}\left (x \,{\mathrm e}^{-b y-a}\right )} = 0
\]
✓ Mathematica. Time used: 0.112 (sec). Leaf size: 380
ode=Log[D[y[x],x]]+x*D[y[x],x]+ a +b*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{y(x)}-\frac {W\left (e^{-a-b K[2]} x\right ) \int _1^x\left (\frac {b^3 W\left (e^{-a-b K[2]} K[1]\right )^2}{K[1] \left (W\left (e^{-a-b K[2]} K[1]\right )+1\right ) \left (b W\left (e^{-a-b K[2]} K[1]\right )-1\right )^2}-\frac {b^2 W\left (e^{-a-b K[2]} K[1]\right )}{K[1] \left (W\left (e^{-a-b K[2]} K[1]\right )+1\right ) \left (b W\left (e^{-a-b K[2]} K[1]\right )-1\right )}\right )dK[1] b+b-\int _1^x\left (\frac {b^3 W\left (e^{-a-b K[2]} K[1]\right )^2}{K[1] \left (W\left (e^{-a-b K[2]} K[1]\right )+1\right ) \left (b W\left (e^{-a-b K[2]} K[1]\right )-1\right )^2}-\frac {b^2 W\left (e^{-a-b K[2]} K[1]\right )}{K[1] \left (W\left (e^{-a-b K[2]} K[1]\right )+1\right ) \left (b W\left (e^{-a-b K[2]} K[1]\right )-1\right )}\right )dK[1]}{b W\left (e^{-a-b K[2]} x\right )-1}dK[2]+\int _1^x\frac {b W\left (e^{-a-b y(x)} K[1]\right )}{K[1] \left (b W\left (e^{-a-b y(x)} K[1]\right )-1\right )}dK[1]=c_1,y(x)\right ]
\]
✓ Sympy. Time used: 1.861 (sec). Leaf size: 97
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(a + b*y(x) + x*Derivative(y(x), x) + log(Derivative(y(x), x)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
C_{1} - b \left (\begin {cases} - y{\left (x \right )} + \frac {\log {\left (x \right )}}{b} - \frac {\log {\left (W\left (e^{- a} e^{- b \left (y{\left (x \right )} - \frac {\log {\left (x \right )}}{b}\right )}\right ) - \frac {1}{b} \right )}}{b} - \frac {W\left (e^{- a} e^{- b \left (y{\left (x \right )} - \frac {\log {\left (x \right )}}{b}\right )}\right )}{b} - \frac {\log {\left (W\left (e^{- a} e^{- b \left (y{\left (x \right )} - \frac {\log {\left (x \right )}}{b}\right )}\right ) - \frac {1}{b} \right )}}{b^{2}} & \text {for}\: b \neq 0 \\- y{\left (x \right )} + \frac {\log {\left (x \right )}}{b} & \text {otherwise} \end {cases}\right ) + \log {\left (x \right )} = 0
\]