83.22.21 problem 21
Internal
problem
ID
[19204]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
IV.
Equations
of
the
first
order
but
not
of
the
first
degree.
Exercise
IV
(E)
at
page
63
Problem
number
:
21
Date
solved
:
Monday, March 31, 2025 at 06:56:39 PM
CAS
classification
:
[_quadrature]
\begin{align*} y&=a y^{\prime }+b {y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.633 (sec). Leaf size: 214
ode:=y(x) = a*diff(y(x),x)+b*diff(y(x),x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\frac {{\mathrm e}^{\frac {-a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {-c_1 -a +x}{a}}}{a \sqrt {\frac {1}{b}}}\right )-a +x -c_1}{a}} a}{\sqrt {\frac {1}{b}}}+b \,{\mathrm e}^{\frac {-2 a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {-c_1 -a +x}{a}}}{a \sqrt {\frac {1}{b}}}\right )-2 a +2 x -2 c_1}{a}}}{b} \\
y &= \frac {a^{2} \left (\operatorname {LambertW}\left (-\frac {2 \sqrt {b}\, {\mathrm e}^{\frac {-c_1 -a +x}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (-\frac {2 \sqrt {b}\, {\mathrm e}^{\frac {-c_1 -a +x}{a}}}{a}\right )}{4 b} \\
y &= \frac {a^{2} \left (\operatorname {LambertW}\left (\frac {2 \sqrt {b}\, {\mathrm e}^{\frac {-c_1 -a +x}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (\frac {2 \sqrt {b}\, {\mathrm e}^{\frac {-c_1 -a +x}{a}}}{a}\right )}{4 b} \\
\end{align*}
✓ Mathematica. Time used: 0.841 (sec). Leaf size: 123
ode=y[x]==a*D[y[x],x]+b*D[y[x],x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {4 \text {$\#$1} b+a^2}+a \log \left (b \left (\sqrt {4 \text {$\#$1} b+a^2}-a\right )\right )}{2 b}\&\right ]\left [\frac {x}{2 b}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {4 \text {$\#$1} b+a^2}-a \log \left (\sqrt {4 \text {$\#$1} b+a^2}+a\right )}{2 b}\&\right ]\left [-\frac {x}{2 b}+c_1\right ] \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 1.232 (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) - b*Derivative(y(x), x)**2 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} - \frac {a \log {\left (a + \sqrt {a^{2} + 4 b y{\left (x \right )}} \right )}}{2 b} + \frac {\sqrt {a^{2} + 4 b y{\left (x \right )}}}{2 b} & \text {for}\: b \neq 0 \\\frac {y{\left (x \right )}}{a + \sqrt {a^{2}}} & \text {otherwise} \end {cases} = C_{1} - \frac {x}{2 b}, \ \frac {- a \log {\left (- a + \sqrt {a^{2} + 4 b y{\left (x \right )}} \right )} + x - \sqrt {a^{2} + 4 b y{\left (x \right )}}}{b} = C_{1}\right ]
\]