29.32.8 problem 942
Internal
problem
ID
[5520]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
32
Problem
number
:
942
Date
solved
:
Sunday, March 30, 2025 at 08:29:20 AM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} y {y^{\prime }}^{2}&=a^{2} x \end{align*}
✓ Maple. Time used: 0.035 (sec). Leaf size: 78
ode:=y(x)*diff(y(x),x)^2 = a^2*x;
dsolve(ode,y(x), singsol=all);
\begin{align*}
x \left (1-\frac {c_1}{{\left (\frac {\left (a \sqrt {y x}\, x -y^{2}\right ) a^{2}}{y^{2}}\right )}^{{2}/{3}} y}\right ) &= 0 \\
x \left (1-\frac {c_1}{\left (-\frac {a^{2} \left (a \sqrt {y x}\, x +y^{2}\right )}{y^{2}}\right )^{{2}/{3}} y}\right ) &= 0 \\
\end{align*}
✓ Mathematica. Time used: 3.878 (sec). Leaf size: 46
ode=y[x] (D[y[x],x])^2==a^2 x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \left (-a x^{3/2}+\frac {3 c_1}{2}\right ){}^{2/3} \\
y(x)\to \left (a x^{3/2}+\frac {3 c_1}{2}\right ){}^{2/3} \\
\end{align*}
✓ Sympy. Time used: 47.890 (sec). Leaf size: 415
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**2*x + y(x)*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} - 2 C_{1} a \sqrt {x^{3}} + a^{2} x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} - 2 C_{1} a \sqrt {x^{3}} + a^{2} x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} + 2 C_{1} a \sqrt {x^{3}} + a^{2} x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} + 2 C_{1} a \sqrt {x^{3}} + a^{2} x^{3}}}{2}, \ y{\left (x \right )} = \sqrt [3]{C_{1}^{2} - 2 C_{1} a \sqrt {x^{3}} + a^{2} x^{3}}, \ y{\left (x \right )} = \sqrt [3]{C_{1}^{2} + 2 C_{1} a \sqrt {x^{3}} + a^{2} x^{3}}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} - 2 C_{1} a \sqrt {x^{3}} + a^{2} x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} - 2 C_{1} a \sqrt {x^{3}} + a^{2} x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} + 2 C_{1} a \sqrt {x^{3}} + a^{2} x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} + 2 C_{1} a \sqrt {x^{3}} + a^{2} x^{3}}}{2}, \ y{\left (x \right )} = \sqrt [3]{C_{1}^{2} - 2 C_{1} a \sqrt {x^{3}} + a^{2} x^{3}}, \ y{\left (x \right )} = \sqrt [3]{C_{1}^{2} + 2 C_{1} a \sqrt {x^{3}} + a^{2} x^{3}}\right ]
\]