23.1.410 problem 397
Internal
problem
ID
[5017]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
397
Date
solved
:
Tuesday, September 30, 2025 at 11:23:15 AM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime } \sqrt {b^{2}-x^{2}}&=\sqrt {a^{2}-y^{2}} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 37
ode:=diff(y(x),x)*(b^2-x^2)^(1/2) = (a^2-y(x)^2)^(1/2);
dsolve(ode,y(x), singsol=all);
\[
\arctan \left (\frac {x}{\sqrt {b^{2}-x^{2}}}\right )-\arctan \left (\frac {y}{\sqrt {a^{2}-y^{2}}}\right )+c_1 = 0
\]
✓ Mathematica. Time used: 2.126 (sec). Leaf size: 66
ode=D[y[x],x]*Sqrt[b^2-x^2]==Sqrt[a^2-y[x]^2];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {a \tan \left (\arctan \left (\frac {x}{\sqrt {b^2-x^2}}\right )+c_1\right )}{\sqrt {\sec ^2\left (\arctan \left (\frac {x}{\sqrt {b^2-x^2}}\right )+c_1\right )}}\\ y(x)&\to -a\\ y(x)&\to a \end{align*}
✓ Sympy. Time used: 8.253 (sec). Leaf size: 335
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-sqrt(a**2 - y(x)**2) + sqrt(b**2 - x**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \begin {cases} \frac {4 C_{1} b^{2}}{- 8 x e^{C_{1}} + 8 i \sqrt {b^{2} - x^{2}} e^{C_{1}}} - \frac {8 C_{1} x^{2}}{- 8 x e^{C_{1}} + 8 i \sqrt {b^{2} - x^{2}} e^{C_{1}}} + \frac {8 i C_{1} x \sqrt {b^{2} - x^{2}}}{- 8 x e^{C_{1}} + 8 i \sqrt {b^{2} - x^{2}} e^{C_{1}}} - \frac {a^{2}}{- 2 x e^{C_{1}} + 2 i \sqrt {b^{2} - x^{2}} e^{C_{1}}} & \text {for}\: a^{2} \neq 0 \wedge b^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} - a^{2} e^{- i C_{1} - \frac {i x \log {\left (x \right )}}{\sqrt {- x^{2}}}} - \frac {e^{i C_{1} + \frac {i x \log {\left (x \right )}}{\sqrt {- x^{2}}}}}{4} & \text {for}\: b^{2} = 0 \wedge a^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \frac {1}{- 2 C_{1} x + 2 i C_{1} \sqrt {b^{2} - x^{2}}} & \text {for}\: a^{2} = 0 \wedge b^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} - 2 C_{1} x + 2 i C_{1} \sqrt {b^{2} - x^{2}} & \text {for}\: a^{2} = 0 \wedge b^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} e^{- \sqrt {- C_{1}^{2} + \frac {2 C_{1} \sqrt {- x^{2}} \log {\left (x \right )}}{x} + \log {\left (x \right )}^{2}}} & \text {for}\: a^{2} = 0 \wedge b^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} e^{\sqrt {- C_{1}^{2} + \frac {2 C_{1} \sqrt {- x^{2}} \log {\left (x \right )}}{x} + \log {\left (x \right )}^{2}}} & \text {for}\: a^{2} = 0 \wedge b^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ]
\]