38.5.75 problem 64
Internal
problem
ID
[8423]
Book
:
A
First
Course
in
Differential
Equations
with
Modeling
Applications
by
Dennis
G.
Zill.
12
ed.
Metric
version.
2024.
Cengage
learning.
Section
:
Chapter
2.
First
order
differential
equations.
Section
2.2
Separable
equations.
Exercises
2.2
at
page
53
Problem
number
:
64
Date
solved
:
Tuesday, September 30, 2025 at 05:36:23 PM
CAS
classification
:
[_quadrature]
\begin{align*} 1+{x^{\prime }}^{2}&=\frac {a}{y} \end{align*}
✓ Maple. Time used: 0.030 (sec). Leaf size: 71
ode:=1+diff(x(y),y)^2 = a/y;
dsolve(ode,x(y), singsol=all);
\begin{align*}
x &= \sqrt {y \left (a -y \right )}-\frac {a \arctan \left (\frac {a -2 y}{2 \sqrt {y \left (a -y \right )}}\right )}{2}+c_1 \\
x &= -\sqrt {y \left (a -y \right )}+\frac {a \arctan \left (\frac {a -2 y}{2 \sqrt {y \left (a -y \right )}}\right )}{2}+c_1 \\
\end{align*}
✓ Mathematica. Time used: 0.205 (sec). Leaf size: 108
ode=1+D[x[y],y]^2== a/y;
ic={};
DSolve[{ode,ic},x[y],y,IncludeSingularSolutions->True]
\begin{align*} x(y)&\to -\sqrt {a-y} \left (\frac {\sqrt {a} \arcsin \left (\frac {\sqrt {y}}{\sqrt {a}}\right )}{\sqrt {1-\frac {y}{a}}}+\sqrt {y}\right )+c_1\\ x(y)&\to \sqrt {a-y} \left (\frac {\sqrt {a} \arcsin \left (\frac {\sqrt {y}}{\sqrt {a}}\right )}{\sqrt {1-\frac {y}{a}}}+\sqrt {y}\right )+c_1 \end{align*}
✓ Sympy. Time used: 1.029 (sec). Leaf size: 190
from sympy import *
y = symbols("y")
a = symbols("a")
x = Function("x")
ode = Eq(-a/y + Derivative(x(y), y)**2 + 1,0)
ics = {}
dsolve(ode,func=x(y),ics=ics)
\[
\left [ x{\left (y \right )} = C_{1} - \begin {cases} - \frac {i \sqrt {a} \sqrt {y}}{\sqrt {-1 + \frac {y}{a}}} - i a \operatorname {acosh}{\left (\frac {\sqrt {y}}{\sqrt {a}} \right )} + \frac {i y^{\frac {3}{2}}}{\sqrt {a} \sqrt {-1 + \frac {y}{a}}} & \text {for}\: \left |{\frac {y}{a}}\right | > 1 \\\sqrt {a} \sqrt {y} \sqrt {1 - \frac {y}{a}} + a \operatorname {asin}{\left (\frac {\sqrt {y}}{\sqrt {a}} \right )} & \text {otherwise} \end {cases}, \ x{\left (y \right )} = C_{1} + \begin {cases} - \frac {i \sqrt {a} \sqrt {y}}{\sqrt {-1 + \frac {y}{a}}} - i a \operatorname {acosh}{\left (\frac {\sqrt {y}}{\sqrt {a}} \right )} + \frac {i y^{\frac {3}{2}}}{\sqrt {a} \sqrt {-1 + \frac {y}{a}}} & \text {for}\: \left |{\frac {y}{a}}\right | > 1 \\\sqrt {a} \sqrt {y} \sqrt {1 - \frac {y}{a}} + a \operatorname {asin}{\left (\frac {\sqrt {y}}{\sqrt {a}} \right )} & \text {otherwise} \end {cases}\right ]
\]