29.32.7 problem 941
Internal
problem
ID
[5519]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
32
Problem
number
:
941
Date
solved
:
Sunday, March 30, 2025 at 08:29:19 AM
CAS
classification
:
[_quadrature]
\begin{align*} y {y^{\prime }}^{2}&=a \end{align*}
✓ Maple. Time used: 0.032 (sec). Leaf size: 169
ode:=y(x)*diff(y(x),x)^2 = a;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {12^{{2}/{3}} \left (a^{2} \left (x -c_1 \right )\right )^{{2}/{3}}}{4 a} \\
y &= \frac {12^{{2}/{3}} \left (a^{2} \left (x -c_1 \right )\right )^{{2}/{3}} \left (1+i \sqrt {3}\right )^{2}}{16 a} \\
y &= \frac {12^{{2}/{3}} \left (a^{2} \left (x -c_1 \right )\right )^{{2}/{3}} \left (i \sqrt {3}-1\right )^{2}}{16 a} \\
y &= \frac {12^{{2}/{3}} \left (a^{2} \left (c_1 -x \right )\right )^{{2}/{3}}}{4 a} \\
y &= \frac {12^{{2}/{3}} \left (a^{2} \left (c_1 -x \right )\right )^{{2}/{3}} \left (1+i \sqrt {3}\right )^{2}}{16 a} \\
y &= \frac {12^{{2}/{3}} \left (a^{2} \left (c_1 -x \right )\right )^{{2}/{3}} \left (i \sqrt {3}-1\right )^{2}}{16 a} \\
\end{align*}
✓ Mathematica. Time used: 3.826 (sec). Leaf size: 54
ode=y[x] (D[y[x],x])^2==a;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \left (\frac {3}{2}\right )^{2/3} \left (-\sqrt {a} x+c_1\right ){}^{2/3} \\
y(x)\to \left (\frac {3}{2}\right )^{2/3} \left (\sqrt {a} x+c_1\right ){}^{2/3} \\
\end{align*}
✓ Sympy. Time used: 18.797 (sec). Leaf size: 190
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a + y(x)*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\sqrt [3]{2} \cdot 3^{\frac {2}{3}} \sqrt [3]{a} \left (C_{1} + x\right )^{\frac {2}{3}}}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \sqrt [3]{a} \left (- 3^{\frac {2}{3}} + 3 \sqrt [6]{3} i\right ) \left (C_{1} + x\right )^{\frac {2}{3}}}{4}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \sqrt [3]{a} \left (- 3^{\frac {2}{3}} - 3 \sqrt [6]{3} i\right ) \left (C_{1} + x\right )^{\frac {2}{3}}}{4}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \cdot 3^{\frac {2}{3}} \sqrt [3]{a} \left (C_{1} + x\right )^{\frac {2}{3}}}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \sqrt [3]{a} \left (- 3^{\frac {2}{3}} + 3 \sqrt [6]{3} i\right ) \left (C_{1} + x\right )^{\frac {2}{3}}}{4}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \sqrt [3]{a} \left (- 3^{\frac {2}{3}} - 3 \sqrt [6]{3} i\right ) \left (C_{1} + x\right )^{\frac {2}{3}}}{4}\right ]
\]