60.1.367 problem 376
Internal
problem
ID
[10381]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
376
Date
solved
:
Sunday, March 30, 2025 at 04:34:12 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{2}+a y^{\prime }+b y&=0 \end{align*}
✓ Maple. Time used: 0.035 (sec). Leaf size: 254
ode:=diff(y(x),x)^2+a*diff(y(x),x)+b*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {a^{2} \left (\operatorname {LambertW}\left (-\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {-a +\left (c_1 -x \right ) b}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (-\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {-a +\left (c_1 -x \right ) b}{a}}}{a}\right )}{4 b} \\
y &= -\frac {a^{2} \left (\operatorname {LambertW}\left (\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {-a +\left (c_1 -x \right ) b}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {-a +\left (c_1 -x \right ) b}{a}}}{a}\right )}{4 b} \\
y &= -\frac {\frac {{\mathrm e}^{\frac {-a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {-a +\left (c_1 -x \right ) b}{a}}}{a \sqrt {-\frac {1}{b}}}\right )-a +\left (c_1 -x \right ) b}{a}} a}{\sqrt {-\frac {1}{b}}}-b \,{\mathrm e}^{\frac {-2 a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {-a +\left (c_1 -x \right ) b}{a}}}{a \sqrt {-\frac {1}{b}}}\right )-2 a +\left (-2 x +2 c_1 \right ) b}{a}}}{b} \\
\end{align*}
✓ Mathematica. Time used: 0.817 (sec). Leaf size: 119
ode=b*y[x] + a*D[y[x],x] + D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {a^2-4 \text {$\#$1} b}+a \log \left (b \left (\sqrt {a^2-4 \text {$\#$1} b}-a\right )\right )}{2 b}\&\right ]\left [\frac {x}{2}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {a^2-4 \text {$\#$1} b}-a \log \left (b \left (\sqrt {a^2-4 \text {$\#$1} b}+a\right )\right )}{2 b}\&\right ]\left [-\frac {x}{2}+c_1\right ] \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 1.200 (sec). Leaf size: 97
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(a*Derivative(y(x), x) + b*y(x) + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} \frac {a \log {\left (a + \sqrt {a^{2} - 4 b y{\left (x \right )}} \right )}}{2 b} - \frac {\sqrt {a^{2} - 4 b y{\left (x \right )}}}{2 b} & \text {for}\: b \neq 0 \\\frac {y{\left (x \right )}}{a + \sqrt {a^{2}}} & \text {otherwise} \end {cases} = C_{1} - \frac {x}{2}, \ \frac {a \log {\left (- a + \sqrt {a^{2} - 4 b y{\left (x \right )}} \right )}}{2 b} + \frac {x}{2} + \frac {\sqrt {a^{2} - 4 b y{\left (x \right )}}}{2 b} = C_{1}\right ]
\]