67.2.24 problem Problem 3(b)
Internal
problem
ID
[13902]
Book
:
APPLIED
DIFFERENTIAL
EQUATIONS
The
Primary
Course
by
Vladimir
A.
Dobrushkin.
CRC
Press
2015
Section
:
Chapter
4,
Second
and
Higher
Order
Linear
Differential
Equations.
Problems
page
221
Problem
number
:
Problem
3(b)
Date
solved
:
Wednesday, March 05, 2025 at 10:21:49 PM
CAS
classification
:
[[_3rd_order, _missing_x]]
\begin{align*} y^{\prime \prime \prime }-5 y^{\prime \prime }+y^{\prime }-y&=0 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 183
ode:=diff(diff(diff(y(x),x),x),x)-5*diff(diff(y(x),x),x)+diff(y(x),x)-y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_{1} {\mathrm e}^{\frac {\left (\left (116+6 \sqrt {78}\right )^{{2}/{3}}+5 \left (116+6 \sqrt {78}\right )^{{1}/{3}}+22\right ) x}{3 \left (116+6 \sqrt {78}\right )^{{1}/{3}}}}-c_{2} {\mathrm e}^{-\frac {\left (22+\left (116+6 \sqrt {78}\right )^{{2}/{3}}-10 \left (116+6 \sqrt {78}\right )^{{1}/{3}}\right ) x}{6 \left (116+6 \sqrt {78}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (116+6 \sqrt {3}\, \sqrt {26}\right )^{{2}/{3}}-22\right ) x}{6 \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{{1}/{3}}}\right )+c_{3} {\mathrm e}^{-\frac {\left (22+\left (116+6 \sqrt {78}\right )^{{2}/{3}}-10 \left (116+6 \sqrt {78}\right )^{{1}/{3}}\right ) x}{6 \left (116+6 \sqrt {78}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (116+6 \sqrt {3}\, \sqrt {26}\right )^{{2}/{3}}-22\right ) x}{6 \left (116+6 \sqrt {3}\, \sqrt {26}\right )^{{1}/{3}}}\right )
\]
✓ Mathematica. Time used: 0.003 (sec). Leaf size: 81
ode=D[y[x],{x,3}]-5*D[y[x],{x,2}]+D[y[x],x]-y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]\right )+c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]\right )
\]
✓ Sympy. Time used: 0.427 (sec). Leaf size: 226
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-y(x) + Derivative(y(x), x) - 5*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} e^{\frac {x \left (- \sqrt [3]{2} \sqrt [3]{3 \sqrt {78} + 58} - \frac {11 \cdot 2^{\frac {2}{3}}}{\sqrt [3]{3 \sqrt {78} + 58}} + 10\right )}{6}} \sin {\left (\frac {\sqrt [3]{2} \sqrt {3} x \left (- \sqrt [3]{3 \sqrt {78} + 58} + \frac {11 \sqrt [3]{2}}{\sqrt [3]{3 \sqrt {78} + 58}}\right )}{6} \right )} + C_{2} e^{\frac {x \left (- \sqrt [3]{2} \sqrt [3]{3 \sqrt {78} + 58} - \frac {11 \cdot 2^{\frac {2}{3}}}{\sqrt [3]{3 \sqrt {78} + 58}} + 10\right )}{6}} \cos {\left (\frac {\sqrt [3]{2} \sqrt {3} x \left (- \sqrt [3]{3 \sqrt {78} + 58} + \frac {11 \sqrt [3]{2}}{\sqrt [3]{3 \sqrt {78} + 58}}\right )}{6} \right )} + C_{3} e^{\frac {x \left (\frac {11 \cdot 2^{\frac {2}{3}}}{\sqrt [3]{3 \sqrt {78} + 58}} + 5 + \sqrt [3]{2} \sqrt [3]{3 \sqrt {78} + 58}\right )}{3}}
\]