15.10.17 problem 17
Internal
problem
ID
[3104]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
18,
page
82
Problem
number
:
17
Date
solved
:
Sunday, March 30, 2025 at 01:18:32 AM
CAS
classification
:
[[_high_order, _missing_x]]
\begin{align*} y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }-6 y^{\prime }+2 y&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 32
ode:=diff(diff(diff(diff(y(x),x),x),x),x)-2*diff(diff(diff(y(x),x),x),x)-6*diff(y(x),x)+2*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \moverset {4}{\munderset {\textit {\_a} =1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =\textit {\_a} \right ) x} \textit {\_C}_{\textit {\_a}}
\]
✓ Mathematica. Time used: 0.003 (sec). Leaf size: 114
ode=D[y[x],{x,4}]-2*D[y[x],{x,3}]-6*D[y[x],x]+2*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^4-2 \text {$\#$1}^3-6 \text {$\#$1}+2\&,1\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^4-2 \text {$\#$1}^3-6 \text {$\#$1}+2\&,3\right ]\right )+c_4 \exp \left (x \text {Root}\left [\text {$\#$1}^4-2 \text {$\#$1}^3-6 \text {$\#$1}+2\&,4\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^4-2 \text {$\#$1}^3-6 \text {$\#$1}+2\&,2\right ]\right )
\]
✓ Sympy. Time used: 1.922 (sec). Leaf size: 583
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*y(x) - 6*Derivative(y(x), x) - 2*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} e^{\frac {x \left (1 - \sqrt {- \frac {2 \cdot 6^{\frac {2}{3}}}{3 \sqrt [3]{99 + 7 \sqrt {201}}} + 1 + \frac {\sqrt [3]{6} \sqrt [3]{99 + 7 \sqrt {201}}}{3}}\right )}{2}} \sin {\left (\frac {x \sqrt {-2 - \frac {2 \cdot 6^{\frac {2}{3}}}{3 \sqrt [3]{99 + 7 \sqrt {201}}} + \frac {\sqrt [3]{6} \sqrt [3]{99 + 7 \sqrt {201}}}{3} + \frac {14}{\sqrt {- \frac {2 \cdot 6^{\frac {2}{3}}}{3 \sqrt [3]{99 + 7 \sqrt {201}}} + 1 + \frac {\sqrt [3]{6} \sqrt [3]{99 + 7 \sqrt {201}}}{3}}}}}{2} \right )} + C_{2} e^{\frac {x \left (1 - \sqrt {- \frac {2 \cdot 6^{\frac {2}{3}}}{3 \sqrt [3]{99 + 7 \sqrt {201}}} + 1 + \frac {\sqrt [3]{6} \sqrt [3]{99 + 7 \sqrt {201}}}{3}}\right )}{2}} \cos {\left (\frac {x \sqrt {-2 - \frac {2 \cdot 6^{\frac {2}{3}}}{3 \sqrt [3]{99 + 7 \sqrt {201}}} + \frac {\sqrt [3]{6} \sqrt [3]{99 + 7 \sqrt {201}}}{3} + \frac {14}{\sqrt {- \frac {2 \cdot 6^{\frac {2}{3}}}{3 \sqrt [3]{99 + 7 \sqrt {201}}} + 1 + \frac {\sqrt [3]{6} \sqrt [3]{99 + 7 \sqrt {201}}}{3}}}}}{2} \right )} + C_{3} e^{\frac {x \left (1 + \sqrt {- \frac {2 \cdot 6^{\frac {2}{3}}}{3 \sqrt [3]{99 + 7 \sqrt {201}}} + 1 + \frac {\sqrt [3]{6} \sqrt [3]{99 + 7 \sqrt {201}}}{3}} + \sqrt {- \frac {\sqrt [3]{6} \sqrt [3]{99 + 7 \sqrt {201}}}{3} + \frac {2 \cdot 6^{\frac {2}{3}}}{3 \sqrt [3]{99 + 7 \sqrt {201}}} + 2 + \frac {14}{\sqrt {- \frac {2 \cdot 6^{\frac {2}{3}}}{3 \sqrt [3]{99 + 7 \sqrt {201}}} + 1 + \frac {\sqrt [3]{6} \sqrt [3]{99 + 7 \sqrt {201}}}{3}}}}\right )}{2}} + C_{4} e^{\frac {x \left (- \sqrt {- \frac {\sqrt [3]{6} \sqrt [3]{99 + 7 \sqrt {201}}}{3} + \frac {2 \cdot 6^{\frac {2}{3}}}{3 \sqrt [3]{99 + 7 \sqrt {201}}} + 2 + \frac {14}{\sqrt {- \frac {2 \cdot 6^{\frac {2}{3}}}{3 \sqrt [3]{99 + 7 \sqrt {201}}} + 1 + \frac {\sqrt [3]{6} \sqrt [3]{99 + 7 \sqrt {201}}}{3}}}} + 1 + \sqrt {- \frac {2 \cdot 6^{\frac {2}{3}}}{3 \sqrt [3]{99 + 7 \sqrt {201}}} + 1 + \frac {\sqrt [3]{6} \sqrt [3]{99 + 7 \sqrt {201}}}{3}}\right )}{2}}
\]