17.3.2 problem 7(b)
Internal
problem
ID
[4143]
Book
:
Theory
and
solutions
of
Ordinary
Differential
equations,
Donald
Greenspan,
1960
Section
:
Chapter
4.
The
general
linear
differential
equation
of
order
n.
Exercises
at
page
63
Problem
number
:
7(b)
Date
solved
:
Tuesday, September 30, 2025 at 07:06:02 AM
CAS
classification
:
[[_3rd_order, _missing_x]]
\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }-12 y&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 117
ode:=diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)-12*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \left (c_1 \,{\mathrm e}^{\frac {x \left (1+\left (163+18 \sqrt {82}\right )^{{2}/{3}}\right )}{2 \left (163+18 \sqrt {82}\right )^{{1}/{3}}}}-c_2 \sin \left (\frac {\sqrt {3}\, \left (\left (163+18 \sqrt {82}\right )^{{2}/{3}}-1\right ) x}{6 \left (163+18 \sqrt {82}\right )^{{1}/{3}}}\right )+c_3 \cos \left (\frac {\sqrt {3}\, \left (\left (163+18 \sqrt {82}\right )^{{2}/{3}}-1\right ) x}{6 \left (163+18 \sqrt {82}\right )^{{1}/{3}}}\right )\right ) {\mathrm e}^{-\frac {\left (\left (163+18 \sqrt {82}\right )^{{1}/{3}}-1\right )^{2} x}{6 \left (163+18 \sqrt {82}\right )^{{1}/{3}}}}
\]
✓ Mathematica. Time used: 0.002 (sec). Leaf size: 75
ode=D[y[x],{x,3}]-D[y[x],{x,2}]-12*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^3-\text {$\#$1}^2-12\&,2\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^3-\text {$\#$1}^2-12\&,3\right ]\right )+c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^3-\text {$\#$1}^2-12\&,1\right ]\right ) \end{align*}
✓ Sympy. Time used: 0.298 (sec). Leaf size: 175
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-12*y(x) - Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} e^{\frac {x \left (- \sqrt [3]{18 \sqrt {82} + 163} - \frac {1}{\sqrt [3]{18 \sqrt {82} + 163}} + 2\right )}{6}} \sin {\left (\frac {\sqrt {3} x \left (- \sqrt [3]{18 \sqrt {82} + 163} + \frac {1}{\sqrt [3]{18 \sqrt {82} + 163}}\right )}{6} \right )} + C_{2} e^{\frac {x \left (- \sqrt [3]{18 \sqrt {82} + 163} - \frac {1}{\sqrt [3]{18 \sqrt {82} + 163}} + 2\right )}{6}} \cos {\left (\frac {\sqrt {3} x \left (- \sqrt [3]{18 \sqrt {82} + 163} + \frac {1}{\sqrt [3]{18 \sqrt {82} + 163}}\right )}{6} \right )} + C_{3} e^{\frac {x \left (\frac {1}{\sqrt [3]{18 \sqrt {82} + 163}} + 1 + \sqrt [3]{18 \sqrt {82} + 163}\right )}{3}}
\]