25.6.10 problem 10
Internal
problem
ID
[6894]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
7
Problem
number
:
10
Date
solved
:
Tuesday, September 30, 2025 at 04:00:24 PM
CAS
classification
:
[_quadrature]
\begin{align*} x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2}&=0 \end{align*}
✓ Maple. Time used: 0.120 (sec). Leaf size: 581
ode:=x^2*(1+diff(y(x),x)^2)^3-a^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 1.711 (sec). Leaf size: 216
ode=x^2*(1+(D[y[x],x])^2)^3-a^2==0;
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: 2.895 (sec). Leaf size: 221
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**2 + x**2*(Derivative(y(x), x)**2 + 1)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} - \int \sqrt {\sqrt [3]{\frac {a^{2}}{x^{2}}} - 1}\, dx, \ y{\left (x \right )} = C_{1} + \int \sqrt {\sqrt [3]{\frac {a^{2}}{x^{2}}} - 1}\, dx, \ y{\left (x \right )} = C_{1} - \frac {\sqrt {2} \int \sqrt {- \sqrt [3]{\frac {a^{2}}{x^{2}}} - \sqrt {3} i \sqrt [3]{\frac {a^{2}}{x^{2}}} - 2}\, dx}{2}, \ y{\left (x \right )} = C_{1} + \frac {\sqrt {2} \int \sqrt {- \sqrt [3]{\frac {a^{2}}{x^{2}}} - \sqrt {3} i \sqrt [3]{\frac {a^{2}}{x^{2}}} - 2}\, dx}{2}, \ y{\left (x \right )} = C_{1} - \frac {\sqrt {2} \int \sqrt {- \sqrt [3]{\frac {a^{2}}{x^{2}}} + \sqrt {3} i \sqrt [3]{\frac {a^{2}}{x^{2}}} - 2}\, dx}{2}, \ y{\left (x \right )} = C_{1} + \frac {\sqrt {2} \int \sqrt {- \sqrt [3]{\frac {a^{2}}{x^{2}}} + \sqrt {3} i \sqrt [3]{\frac {a^{2}}{x^{2}}} - 2}\, dx}{2}\right ]
\]