75.8.17 problem 215
Internal
problem
ID
[16762]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
8.
First
order
not
solved
for
the
derivative.
Exercises
page
67
Problem
number
:
215
Date
solved
:
Monday, March 31, 2025 at 03:16:19 PM
CAS
classification
:
[_quadrature]
\begin{align*} x \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}}&=a \end{align*}
✓ Maple. Time used: 0.352 (sec). Leaf size: 229
ode:=x*(1+diff(y(x),x)^2)^(3/2) = a;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \int \frac {\sqrt {\left (a \,x^{2}\right )^{{2}/{3}}-x^{2}}}{x}d x +c_1 \\
y &= -\frac {\int \frac {\sqrt {-2 i \sqrt {3}\, \left (a \,x^{2}\right )^{{2}/{3}}-2 \left (a \,x^{2}\right )^{{2}/{3}}-4 x^{2}}}{x}d x}{2}+c_1 \\
y &= \frac {\int \frac {\sqrt {-2 i \sqrt {3}\, \left (a \,x^{2}\right )^{{2}/{3}}-2 \left (a \,x^{2}\right )^{{2}/{3}}-4 x^{2}}}{x}d x}{2}+c_1 \\
y &= -\int \frac {\sqrt {\left (a \,x^{2}\right )^{{2}/{3}}-x^{2}}}{x}d x +c_1 \\
y &= -\frac {\sqrt {2}\, \int \frac {\sqrt {i \sqrt {3}\, \left (a \,x^{2}\right )^{{2}/{3}}-\left (a \,x^{2}\right )^{{2}/{3}}-2 x^{2}}}{x}d x}{2}+c_1 \\
y &= \frac {\sqrt {2}\, \int \frac {\sqrt {i \sqrt {3}\, \left (a \,x^{2}\right )^{{2}/{3}}-\left (a \,x^{2}\right )^{{2}/{3}}-2 x^{2}}}{x}d x}{2}+c_1 \\
\end{align*}
✓ Mathematica. Time used: 0.343 (sec). Leaf size: 216
ode=x*(1+D[y[x],x]^2)^(3/2)==a;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -x \left (\frac {a^{2/3}}{x^{2/3}}-1\right )^{3/2}+c_1 \\
y(x)\to x \left (\frac {a^{2/3}}{x^{2/3}}-1\right )^{3/2}+c_1 \\
y(x)\to c_1-x \left (-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2} \\
y(x)\to x \left (-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}+c_1 \\
y(x)\to c_1-x \left (-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2} \\
y(x)\to x \left (-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}+c_1 \\
\end{align*}
✓ Sympy. Time used: 4.307 (sec). Leaf size: 221
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a + x*(Derivative(y(x), x)**2 + 1)**(3/2),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} - \int \sqrt {\sqrt [3]{\frac {a^{2}}{x^{2}}} - 1}\, dx, \ y{\left (x \right )} = C_{1} + \int \sqrt {\sqrt [3]{\frac {a^{2}}{x^{2}}} - 1}\, dx, \ y{\left (x \right )} = C_{1} - \frac {\sqrt {2} \int \sqrt {- \sqrt [3]{\frac {a^{2}}{x^{2}}} - \sqrt {3} i \sqrt [3]{\frac {a^{2}}{x^{2}}} - 2}\, dx}{2}, \ y{\left (x \right )} = C_{1} + \frac {\sqrt {2} \int \sqrt {- \sqrt [3]{\frac {a^{2}}{x^{2}}} - \sqrt {3} i \sqrt [3]{\frac {a^{2}}{x^{2}}} - 2}\, dx}{2}, \ y{\left (x \right )} = C_{1} - \frac {\sqrt {2} \int \sqrt {- \sqrt [3]{\frac {a^{2}}{x^{2}}} + \sqrt {3} i \sqrt [3]{\frac {a^{2}}{x^{2}}} - 2}\, dx}{2}, \ y{\left (x \right )} = C_{1} + \frac {\sqrt {2} \int \sqrt {- \sqrt [3]{\frac {a^{2}}{x^{2}}} + \sqrt {3} i \sqrt [3]{\frac {a^{2}}{x^{2}}} - 2}\, dx}{2}\right ]
\]