60.1.485 problem 498
Internal
problem
ID
[10499]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
498
Date
solved
:
Sunday, March 30, 2025 at 05:20:37 PM
CAS
classification
:
[_quadrature]
\begin{align*} \left (3 y-2\right ) {y^{\prime }}^{2}-4+4 y&=0 \end{align*}
✓ Maple. Time used: 0.040 (sec). Leaf size: 99
ode:=(3*y(x)-2)*diff(y(x),x)^2-4+4*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 1 \\
y &= \frac {\sin \left (\operatorname {RootOf}\left (8 \sqrt {3}\, c_1 \textit {\_Z} -8 \sqrt {3}\, x \textit {\_Z} +\cos \left (\textit {\_Z} \right )^{2}-48 c_1^{2}+96 c_1 x -48 x^{2}-\textit {\_Z}^{2}\right )\right )}{6}+\frac {5}{6} \\
y &= \frac {\sin \left (\operatorname {RootOf}\left (8 \sqrt {3}\, c_1 \textit {\_Z} -8 \sqrt {3}\, x \textit {\_Z} -\cos \left (\textit {\_Z} \right )^{2}+48 c_1^{2}-96 c_1 x +48 x^{2}+\textit {\_Z}^{2}\right )\right )}{6}+\frac {5}{6} \\
\end{align*}
✓ Mathematica. Time used: 0.375 (sec). Leaf size: 160
ode=-4 + 4*y[x] + (-2 + 3*y[x])*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 {1-\text {$\#$1}} \text {arcsinh}\left (\sqrt {3} \sqrt {\text {$\#$1}-1}\right )}{\sqrt {3} \sqrt {\text {$\#$1}-1}}-\sqrt {3 (\text {$\#$1}-1)+1} \sqrt {1-\text {$\#$1}}\&\right ][-2 x+c_1] \\
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}} \text {arcsinh}\left (\sqrt {3} \sqrt {\text {$\#$1}-1}\right )}{\sqrt {3} \sqrt {\text {$\#$1}-1}}-\sqrt {3 (\text {$\#$1}-1)+1} \sqrt {1-\text {$\#$1}}\&\right ][2 x+c_1] \\
y(x)\to 1 \\
\end{align*}
✓ Sympy. Time used: 1.996 (sec). Leaf size: 41
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((3*y(x) - 2)*Derivative(y(x), x)**2 + 4*y(x) - 4,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {\frac {1 - y}{3 y - 2}}}\, dy = C_{1} + 2 x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {\frac {1 - y}{3 y - 2}}}\, dy = C_{1} - 2 x\right ]
\]