83.18.18 problem 18
Internal
problem
ID
[19149]
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
(A)
at
page
53
Problem
number
:
18
Date
solved
:
Thursday, March 13, 2025 at 01:45:46 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{2} \left (2-3 y\right )^{2}&=4-4 y \end{align*}
✓ Maple. Time used: 0.053 (sec). Leaf size: 469
ode:=diff(y(x),x)^2*(2-3*y(x))^2 = 4-4*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= 1 \\
y \left (x \right ) &= -\frac {{\left (\left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{1}/{3}}+\frac {12}{\left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{1}/{3}}}\right )}^{2}}{36}+1 \\
y \left (x \right ) &= 1+\frac {{\left (\left (i-\sqrt {3}\right ) \left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{2}/{3}}+12 i+12 \sqrt {3}\right )}^{2}}{144 \left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{2}/{3}}} \\
y \left (x \right ) &= 1+\frac {{\left (\left (\sqrt {3}+i\right ) \left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{2}/{3}}+12 i-12 \sqrt {3}\right )}^{2}}{144 \left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{2}/{3}}} \\
y \left (x \right ) &= -\frac {{\left (\left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{1}/{3}}+\frac {12}{\left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{1}/{3}}}\right )}^{2}}{36}+1 \\
y \left (x \right ) &= 1+\frac {{\left (\left (i-\sqrt {3}\right ) \left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{2}/{3}}+12 i+12 \sqrt {3}\right )}^{2}}{144 \left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{2}/{3}}} \\
y \left (x \right ) &= 1+\frac {{\left (\left (\sqrt {3}+i\right ) \left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{2}/{3}}+12 i-12 \sqrt {3}\right )}^{2}}{144 \left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{2}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 4.384 (sec). Leaf size: 896
ode=D[y[x],x]^2*(2-3*y[x])^2==4*(1-y[x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Sympy. Time used: 0.941 (sec). Leaf size: 214
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((2 - 3*y(x))**2*Derivative(y(x), x)**2 + 4*y(x) - 4,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} - \frac {2 \sqrt {3} i \left (y{\left (x \right )} - \frac {2}{3}\right ) \sqrt {3 y{\left (x \right )} - 3}}{3} - \frac {4 \sqrt {3} i \sqrt {3 y{\left (x \right )} - 3}}{9} & \text {for}\: \left |{y{\left (x \right )} - \frac {2}{3}}\right | > \frac {1}{3} \\- \frac {2 \sqrt {3} \sqrt {3 - 3 y{\left (x \right )}} \left (y{\left (x \right )} - \frac {2}{3}\right )}{3} - \frac {4 \sqrt {3} \sqrt {3 - 3 y{\left (x \right )}}}{9} & \text {otherwise} \end {cases} = C_{1} - 2 x, \ \begin {cases} - \frac {2 \sqrt {3} i \left (y{\left (x \right )} - \frac {2}{3}\right ) \sqrt {3 y{\left (x \right )} - 3}}{3} - \frac {4 \sqrt {3} i \sqrt {3 y{\left (x \right )} - 3}}{9} & \text {for}\: \left |{y{\left (x \right )} - \frac {2}{3}}\right | > \frac {1}{3} \\- \frac {2 \sqrt {3} \sqrt {3 - 3 y{\left (x \right )}} \left (y{\left (x \right )} - \frac {2}{3}\right )}{3} - \frac {4 \sqrt {3} \sqrt {3 - 3 y{\left (x \right )}}}{9} & \text {otherwise} \end {cases} = C_{1} + 2 x\right ]
\]