7.5.52 problem 52
Internal
problem
ID
[156]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
1.
First
order
differential
equations.
Section
1.6
(substitution
and
exact
equations).
Problems
at
page
72
Problem
number
:
52
Date
solved
:
Saturday, March 29, 2025 at 04:37:18 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y^{3} y^{\prime \prime }&=1 \end{align*}
✓ Maple. Time used: 0.301 (sec). Leaf size: 46
ode:=y(x)^3*diff(diff(y(x),x),x) = 1;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\sqrt {c_1 \left (1+\left (c_2 +x \right )^{2} c_1^{2}\right )}}{c_1} \\
y &= -\frac {\sqrt {c_1 \left (1+\left (c_2 +x \right )^{2} c_1^{2}\right )}}{c_1} \\
\end{align*}
✓ Mathematica. Time used: 3.387 (sec). Leaf size: 93
ode=y[x]^3*D[y[x],{x,2}]==1;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x+1+c_2{}^2 c_1{}^2}}{\sqrt {c_1}} \\
y(x)\to \frac {\sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x+1+c_2{}^2 c_1{}^2}}{\sqrt {c_1}} \\
y(x)\to \text {Indeterminate} \\
\end{align*}
✓ Sympy. Time used: 0.497 (sec). Leaf size: 112
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(y(x)**3*Derivative(y(x), (x, 2)) - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} \frac {i \sqrt {-1 + \frac {1}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {for}\: \frac {1}{\left |{C_{1} y^{2}{\left (x \right )}}\right |} > 1 \\\frac {\sqrt {1 - \frac {1}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {otherwise} \end {cases} = C_{1} + x, \ \begin {cases} \frac {i \sqrt {-1 + \frac {1}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {for}\: \frac {1}{\left |{C_{1} y^{2}{\left (x \right )}}\right |} > 1 \\\frac {\sqrt {1 - \frac {1}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {otherwise} \end {cases} = C_{1} - x\right ]
\]