81.7.4 problem 5
Internal
problem
ID
[18622]
Book
:
A
short
course
on
differential
equations.
By
Donald
Francis
Campbell.
Maxmillan
company.
London.
1907
Section
:
Chapter
VI.
Certain
particular
forms
of
equations.
Exercises
at
page
74
Problem
number
:
5
Date
solved
:
Thursday, March 13, 2025 at 12:25:33 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y^{\prime \prime }&=\frac {1}{y^{2}} \end{align*}
✓ Maple. Time used: 0.020 (sec). Leaf size: 335
ode:=diff(diff(y(x),x),x) = 1/y(x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 0.148 (sec). Leaf size: 62
ode=D[y[x],{x,2}]==1/y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\left (\frac {2 \text {arctanh}\left (\frac {\sqrt {-\frac {2}{y(x)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}+\frac {y(x) \sqrt {-\frac {2}{y(x)}+c_1}}{c_1}\right ){}^2=(x+c_2){}^2,y(x)\right ]
\]
✓ Sympy. Time used: 1.789 (sec). Leaf size: 262
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), (x, 2)) - 1/y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} C_{1} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2} \right )} + \frac {y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {C_{1} y{\left (x \right )} - 2}} - \frac {2 \sqrt {y{\left (x \right )}}}{C_{1} \sqrt {C_{1} y{\left (x \right )} - 2}} & \text {for}\: \left |{C_{1} y{\left (x \right )}}\right | > 2 \\- i C_{1} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2} \right )} - \frac {i y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {- C_{1} y{\left (x \right )} + 2}} + \frac {2 i \sqrt {y{\left (x \right )}}}{C_{1} \sqrt {- C_{1} y{\left (x \right )} + 2}} & \text {otherwise} \end {cases} = C_{1} + x, \ \begin {cases} C_{1} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2} \right )} + \frac {y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {C_{1} y{\left (x \right )} - 2}} - \frac {2 \sqrt {y{\left (x \right )}}}{C_{1} \sqrt {C_{1} y{\left (x \right )} - 2}} & \text {for}\: \left |{C_{1} y{\left (x \right )}}\right | > 2 \\- i C_{1} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2} \right )} - \frac {i y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {- C_{1} y{\left (x \right )} + 2}} + \frac {2 i \sqrt {y{\left (x \right )}}}{C_{1} \sqrt {- C_{1} y{\left (x \right )} + 2}} & \text {otherwise} \end {cases} = C_{1} - x\right ]
\]