82.42.4 problem Ex. 4
Internal
problem
ID
[18881]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
VIII.
Exact
differential
equations,
and
equations
of
particular
forms.
Integration
in
series.
problems
at
page
97
Problem
number
:
Ex.
4
Date
solved
:
Thursday, March 13, 2025 at 01:06:52 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y^{\prime \prime }-\frac {a^{2}}{y^{2}}&=0 \end{align*}
✓ Maple. Time used: 0.020 (sec). Leaf size: 385
ode:=diff(diff(y(x),x),x)-a^2/y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 0.173 (sec). Leaf size: 71
ode=D[y[x],{x,2}]-a^2/y[x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\left (\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {-\frac {2 a^2}{y(x)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}+\frac {y(x) \sqrt {-\frac {2 a^2}{y(x)}+c_1}}{c_1}\right ){}^2=(x+c_2){}^2,y(x)\right ]
\]
✓ Sympy. Time used: 2.237 (sec). Leaf size: 405
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**2/y(x)**2 + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} - \frac {\sqrt {2} C_{1} a \sqrt {y{\left (x \right )}}}{\sqrt {\frac {C_{1} y{\left (x \right )}}{2 a^{2}} - 1}} + \frac {\sqrt {2} y^{\frac {3}{2}}{\left (x \right )}}{2 a \sqrt {\frac {C_{1} y{\left (x \right )}}{2 a^{2}} - 1}} + \frac {2 a^{2} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 a} \right )}}{C_{1}^{\frac {3}{2}}} & \text {for}\: \left |{\frac {C_{1} y{\left (x \right )}}{a^{2}}}\right | > 2 \\\frac {\sqrt {2} i C_{1} a \sqrt {y{\left (x \right )}}}{\sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a^{2}} + 1}} - \frac {\sqrt {2} i y^{\frac {3}{2}}{\left (x \right )}}{2 a \sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a^{2}} + 1}} - \frac {2 i a^{2} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 a} \right )}}{C_{1}^{\frac {3}{2}}} & \text {otherwise} \end {cases} = C_{1} + x, \ \begin {cases} - \frac {\sqrt {2} C_{1} a \sqrt {y{\left (x \right )}}}{\sqrt {\frac {C_{1} y{\left (x \right )}}{2 a^{2}} - 1}} + \frac {\sqrt {2} y^{\frac {3}{2}}{\left (x \right )}}{2 a \sqrt {\frac {C_{1} y{\left (x \right )}}{2 a^{2}} - 1}} + \frac {2 a^{2} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 a} \right )}}{C_{1}^{\frac {3}{2}}} & \text {for}\: \left |{\frac {C_{1} y{\left (x \right )}}{a^{2}}}\right | > 2 \\\frac {\sqrt {2} i C_{1} a \sqrt {y{\left (x \right )}}}{\sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a^{2}} + 1}} - \frac {\sqrt {2} i y^{\frac {3}{2}}{\left (x \right )}}{2 a \sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a^{2}} + 1}} - \frac {2 i a^{2} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 a} \right )}}{C_{1}^{\frac {3}{2}}} & \text {otherwise} \end {cases} = C_{1} - x\right ]
\]