83.22.21 problem 21
Internal
problem
ID
[19275]
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
:
Tuesday, January 28, 2025 at 01:22:05 PM
CAS
classification
:
[_quadrature]
\begin{align*} y&=a y^{\prime }+b {y^{\prime }}^{2} \end{align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 207
dsolve(y(x)=a*diff(y(x),x)+b*diff(y(x),x)^2,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= {\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}} \left (a \sqrt {\frac {1}{b}}+{\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}}\right ) \\
y \left (x \right ) &= \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 \left (x \right ) &= \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*}
✓ Solution by Mathematica
Time used: 0.841 (sec). Leaf size: 123
DSolve[y[x]==a*D[y[x],x]+b*D[y[x],x]^2,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*}