60.1.392 problem 403
Internal
problem
ID
[10406]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
403
Date
solved
:
Wednesday, March 05, 2025 at 10:44:59 AM
CAS
classification
:
[_quadrature]
\begin{align*} a {y^{\prime }}^{2}+b y^{\prime }-y&=0 \end{align*}
✓ Maple. Time used: 0.576 (sec). Leaf size: 207
ode:=a*diff(y(x),x)^2+b*diff(y(x),x)-y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= {\mathrm e}^{\frac {-b \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {-c_{1} -b +x}{b}}}{b \sqrt {\frac {1}{a}}}\right )-b +x -c_{1}}{b}} \left (b \sqrt {\frac {1}{a}}+{\mathrm e}^{\frac {-b \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {-c_{1} -b +x}{b}}}{b \sqrt {\frac {1}{a}}}\right )-b +x -c_{1}}{b}}\right ) \\
y &= \frac {b^{2} \left (\operatorname {LambertW}\left (-\frac {2 \sqrt {a}\, {\mathrm e}^{\frac {-c_{1} -b +x}{b}}}{b}\right )+2\right ) \operatorname {LambertW}\left (-\frac {2 \sqrt {a}\, {\mathrm e}^{\frac {-c_{1} -b +x}{b}}}{b}\right )}{4 a} \\
y &= \frac {b^{2} \left (\operatorname {LambertW}\left (\frac {2 \sqrt {a}\, {\mathrm e}^{\frac {-c_{1} -b +x}{b}}}{b}\right )+2\right ) \operatorname {LambertW}\left (\frac {2 \sqrt {a}\, {\mathrm e}^{\frac {-c_{1} -b +x}{b}}}{b}\right )}{4 a} \\
\end{align*}
✓ Mathematica. Time used: 0.852 (sec). Leaf size: 123
ode=-y[x] + b*D[y[x],x] + a*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 {4 \text {$\#$1} a+b^2}+b \log \left (a \left (\sqrt {4 \text {$\#$1} a+b^2}-b\right )\right )}{2 a}\&\right ]\left [\frac {x}{2 a}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {4 \text {$\#$1} a+b^2}-b \log \left (\sqrt {4 \text {$\#$1} a+b^2}+b\right )}{2 a}\&\right ]\left [-\frac {x}{2 a}+c_1\right ] \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 1.181 (sec). Leaf size: 92
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(a*Derivative(y(x), x)**2 + b*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} - \frac {b \log {\left (b + \sqrt {4 a y{\left (x \right )} + b^{2}} \right )}}{2 a} + \frac {\sqrt {4 a y{\left (x \right )} + b^{2}}}{2 a} & \text {for}\: a \neq 0 \\\frac {y{\left (x \right )}}{b + \sqrt {b^{2}}} & \text {otherwise} \end {cases} = C_{1} - \frac {x}{2 a}, \ \frac {- b \log {\left (- b + \sqrt {4 a y{\left (x \right )} + b^{2}} \right )} + x - \sqrt {4 a y{\left (x \right )} + b^{2}}}{a} = C_{1}\right ]
\]