88.12.8 problem 10
Internal
problem
ID
[24064]
Book
:
Elementary
Differential
Equations.
By
Lee
Roy
Wilcox
and
Herbert
J.
Curtis.
1961
first
edition.
International
texbook
company.
Scranton,
Penn.
USA.
CAT
number
61-15976
Section
:
Chapter
2.
Differential
equations
of
first
order.
Miscellaneous
Exercises
at
page
55
Problem
number
:
10
Date
solved
:
Thursday, October 02, 2025 at 09:56:45 PM
CAS
classification
:
[_separable]
\begin{align*} x \sqrt {a^{2}+x^{2}}&=y \sqrt {y^{2}-a^{2}}\, y^{\prime } \end{align*}
✓ Maple. Time used: 0.001 (sec). Leaf size: 37
ode:=x*(a^2+x^2)^(1/2) = y(x)*(y(x)^2-a^2)^(1/2)*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\[
c_1 +\left (a^{2}+x^{2}\right )^{{3}/{2}}+\left (a -y\right ) \left (y+a \right ) \sqrt {y^{2}-a^{2}} = 0
\]
✓ Mathematica. Time used: 5.878 (sec). Leaf size: 541
ode=x*Sqrt[x^2+a^2]==y[x]*Sqrt[y[x]^2-a^2]*D[y[x],{x,1}];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\sqrt {a^2+\sqrt [3]{a^6+3 a^4 x^2+6 c_1 x^2 \sqrt {a^2+x^2}+3 a^2 \left (x^4+2 c_1 \sqrt {a^2+x^2}\right )+x^6+9 c_1{}^2}}\\ y(x)&\to \sqrt {a^2+\sqrt [3]{a^6+3 a^4 x^2+6 c_1 x^2 \sqrt {a^2+x^2}+3 a^2 \left (x^4+2 c_1 \sqrt {a^2+x^2}\right )+x^6+9 c_1{}^2}}\\ y(x)&\to -\sqrt {a^2-\frac {1}{2} i \left (\sqrt {3}-i\right ) \sqrt [3]{a^6+3 a^4 x^2+6 c_1 x^2 \sqrt {a^2+x^2}+3 a^2 \left (x^4+2 c_1 \sqrt {a^2+x^2}\right )+x^6+9 c_1{}^2}}\\ y(x)&\to \sqrt {a^2-\frac {1}{2} i \left (\sqrt {3}-i\right ) \sqrt [3]{a^6+3 a^4 x^2+6 c_1 x^2 \sqrt {a^2+x^2}+3 a^2 \left (x^4+2 c_1 \sqrt {a^2+x^2}\right )+x^6+9 c_1{}^2}}\\ y(x)&\to -\sqrt {a^2+\frac {1}{2} i \left (\sqrt {3}+i\right ) \sqrt [3]{a^6+3 a^4 x^2+6 c_1 x^2 \sqrt {a^2+x^2}+3 a^2 \left (x^4+2 c_1 \sqrt {a^2+x^2}\right )+x^6+9 c_1{}^2}}\\ y(x)&\to \sqrt {a^2+\frac {1}{2} i \left (\sqrt {3}+i\right ) \sqrt [3]{a^6+3 a^4 x^2+6 c_1 x^2 \sqrt {a^2+x^2}+3 a^2 \left (x^4+2 c_1 \sqrt {a^2+x^2}\right )+x^6+9 c_1{}^2}} \end{align*}
✓ Sympy. Time used: 48.315 (sec). Leaf size: 760
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(x*sqrt(a**2 + x**2) - sqrt(-a**2 + y(x)**2)*y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \sqrt {a^{2} + \sqrt [3]{9 C_{1}^{2} + 6 C_{1} a^{2} \sqrt {a^{2} + x^{2}} + 6 C_{1} x^{2} \sqrt {a^{2} + x^{2}} + a^{6} + 3 a^{4} x^{2} + 3 a^{2} x^{4} + x^{6}}}, \ y{\left (x \right )} = \sqrt {a^{2} + \sqrt [3]{9 C_{1}^{2} + 6 C_{1} a^{2} \sqrt {a^{2} + x^{2}} + 6 C_{1} x^{2} \sqrt {a^{2} + x^{2}} + a^{6} + 3 a^{4} x^{2} + 3 a^{2} x^{4} + x^{6}}}, \ y{\left (x \right )} = - \sqrt {a^{2} - \frac {\sqrt [3]{9 C_{1}^{2} + 6 C_{1} a^{2} \sqrt {a^{2} + x^{2}} + 6 C_{1} x^{2} \sqrt {a^{2} + x^{2}} + a^{6} + 3 a^{4} x^{2} + 3 a^{2} x^{4} + x^{6}}}{2} - \frac {\sqrt {3} i \sqrt [3]{9 C_{1}^{2} + 6 C_{1} a^{2} \sqrt {a^{2} + x^{2}} + 6 C_{1} x^{2} \sqrt {a^{2} + x^{2}} + a^{6} + 3 a^{4} x^{2} + 3 a^{2} x^{4} + x^{6}}}{2}}, \ y{\left (x \right )} = \sqrt {a^{2} - \frac {\sqrt [3]{9 C_{1}^{2} + 6 C_{1} a^{2} \sqrt {a^{2} + x^{2}} + 6 C_{1} x^{2} \sqrt {a^{2} + x^{2}} + a^{6} + 3 a^{4} x^{2} + 3 a^{2} x^{4} + x^{6}}}{2} - \frac {\sqrt {3} i \sqrt [3]{9 C_{1}^{2} + 6 C_{1} a^{2} \sqrt {a^{2} + x^{2}} + 6 C_{1} x^{2} \sqrt {a^{2} + x^{2}} + a^{6} + 3 a^{4} x^{2} + 3 a^{2} x^{4} + x^{6}}}{2}}, \ y{\left (x \right )} = - \sqrt {a^{2} - \frac {\sqrt [3]{9 C_{1}^{2} + 6 C_{1} a^{2} \sqrt {a^{2} + x^{2}} + 6 C_{1} x^{2} \sqrt {a^{2} + x^{2}} + a^{6} + 3 a^{4} x^{2} + 3 a^{2} x^{4} + x^{6}}}{2} + \frac {\sqrt {3} i \sqrt [3]{9 C_{1}^{2} + 6 C_{1} a^{2} \sqrt {a^{2} + x^{2}} + 6 C_{1} x^{2} \sqrt {a^{2} + x^{2}} + a^{6} + 3 a^{4} x^{2} + 3 a^{2} x^{4} + x^{6}}}{2}}, \ y{\left (x \right )} = \sqrt {a^{2} - \frac {\sqrt [3]{9 C_{1}^{2} + 6 C_{1} a^{2} \sqrt {a^{2} + x^{2}} + 6 C_{1} x^{2} \sqrt {a^{2} + x^{2}} + a^{6} + 3 a^{4} x^{2} + 3 a^{2} x^{4} + x^{6}}}{2} + \frac {\sqrt {3} i \sqrt [3]{9 C_{1}^{2} + 6 C_{1} a^{2} \sqrt {a^{2} + x^{2}} + 6 C_{1} x^{2} \sqrt {a^{2} + x^{2}} + a^{6} + 3 a^{4} x^{2} + 3 a^{2} x^{4} + x^{6}}}{2}}\right ]
\]