62.20.4 problem Ex 4
Internal
problem
ID
[12839]
Book
:
An
elementary
treatise
on
differential
equations
by
Abraham
Cohen.
DC
heath
publishers.
1906
Section
:
Chapter
V,
Singular
solutions.
Article
33.
Page
73
Problem
number
:
Ex
4
Date
solved
:
Wednesday, March 05, 2025 at 08:48:11 PM
CAS
classification
:
[_quadrature]
\begin{align*} y \left (3-4 y\right )^{2} {y^{\prime }}^{2}&=4-4 y \end{align*}
✓ Maple. Time used: 0.096 (sec). Leaf size: 58
ode:=y(x)*(3-4*y(x))^2*diff(y(x),x)^2 = 4-4*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 1 \\
x +\frac {y^{2} \left (y-1\right )}{\sqrt {-y \left (y-1\right )}}-c_{1} &= 0 \\
x -\frac {y^{2} \left (y-1\right )}{\sqrt {-y \left (y-1\right )}}-c_{1} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 60.226 (sec). Leaf size: 3751
ode=y[x]*(3-4*y[x])^2*D[y[x],x]^2==4*(1-y[x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Too large to display
✓ Sympy. Time used: 4.917 (sec). Leaf size: 228
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((3 - 4*y(x))**2*y(x)*Derivative(y(x), x)**2 + 4*y(x) - 4,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} 3 i \sqrt {y{\left (x \right )} - 1} \sqrt {y{\left (x \right )}} - \frac {2 i y^{\frac {5}{2}}{\left (x \right )}}{\sqrt {y{\left (x \right )} - 1}} - \frac {i y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {y{\left (x \right )} - 1}} + \frac {3 i \sqrt {y{\left (x \right )}}}{\sqrt {y{\left (x \right )} - 1}} & \text {for}\: \left |{y{\left (x \right )}}\right | > 1 \\\frac {2 y^{\frac {5}{2}}{\left (x \right )}}{\sqrt {1 - y{\left (x \right )}}} - \frac {2 y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {1 - y{\left (x \right )}}} & \text {otherwise} \end {cases} = C_{1} - 2 x, \ \begin {cases} 3 i \sqrt {y{\left (x \right )} - 1} \sqrt {y{\left (x \right )}} - \frac {2 i y^{\frac {5}{2}}{\left (x \right )}}{\sqrt {y{\left (x \right )} - 1}} - \frac {i y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {y{\left (x \right )} - 1}} + \frac {3 i \sqrt {y{\left (x \right )}}}{\sqrt {y{\left (x \right )} - 1}} & \text {for}\: \left |{y{\left (x \right )}}\right | > 1 \\\frac {2 y^{\frac {5}{2}}{\left (x \right )}}{\sqrt {1 - y{\left (x \right )}}} - \frac {2 y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {1 - y{\left (x \right )}}} & \text {otherwise} \end {cases} = C_{1} + 2 x\right ]
\]