84.22.7 problem 13.17
Internal
problem
ID
[22242]
Book
:
Schaums
outline
series.
Differential
Equations
By
Richard
Bronson.
1973.
McGraw-Hill
Inc.
ISBN
0-07-008009-7
Section
:
Chapter
13.
nth
Order
linear
homogeneous
differential
equations
with
constant
coefficients.
Supplementary
problems
Problem
number
:
13.17
Date
solved
:
Thursday, October 02, 2025 at 08:36:32 PM
CAS
classification
:
[[_high_order, _missing_x]]
\begin{align*} y^{\prime \prime \prime \prime }-2 y^{\prime \prime }+16 y^{\prime }+32 y&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 32
ode:=diff(diff(diff(diff(y(x),x),x),x),x)-2*diff(diff(y(x),x),x)+16*diff(y(x),x)+32*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \moverset {4}{\munderset {\textit {\_a} =1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{2}+16 \textit {\_Z} +32, \operatorname {index} =\textit {\_a} \right ) x} \textit {\_C}_{\textit {\_a}}
\]
✓ Mathematica. Time used: 0.002 (sec). Leaf size: 114
ode=D[y[x],{x,4}]-2*D[y[x],{x,2}]+16*D[y[x],{x,1}]+32*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^4-2 \text {$\#$1}^2+16 \text {$\#$1}+32\&,1\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^4-2 \text {$\#$1}^2+16 \text {$\#$1}+32\&,2\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^4-2 \text {$\#$1}^2+16 \text {$\#$1}+32\&,3\right ]\right )+c_4 \exp \left (x \text {Root}\left [\text {$\#$1}^4-2 \text {$\#$1}^2+16 \text {$\#$1}+32\&,4\right ]\right ) \end{align*}
✓ Sympy. Time used: 1.481 (sec). Leaf size: 447
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(32*y(x) + 16*Derivative(y(x), x) - 2*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (C_{1} \sin {\left (\frac {x \sqrt {\left |{- \frac {8}{3} + \frac {194}{3 \sqrt [3]{719 + 24 \sqrt {687} i}} + \frac {16 \sqrt {6}}{\sqrt {2 + \frac {97}{\sqrt [3]{719 + 24 \sqrt {687} i}} + \sqrt [3]{719 + 24 \sqrt {687} i}}} + \frac {2 \sqrt [3]{719 + 24 \sqrt {687} i}}{3}}\right |}}{2} \right )} + C_{2} \cos {\left (\frac {x \sqrt {- \frac {8}{3} + \frac {194}{3 \sqrt [3]{719 + 24 \sqrt {687} i}} + \frac {16 \sqrt {6}}{\sqrt {2 + \frac {97}{\sqrt [3]{719 + 24 \sqrt {687} i}} + \sqrt [3]{719 + 24 \sqrt {687} i}}} + \frac {2 \sqrt [3]{719 + 24 \sqrt {687} i}}{3}}}{2} \right )}\right ) e^{\frac {\sqrt {6} x \sqrt {2 + \frac {97}{\sqrt [3]{719 + 24 \sqrt {687} i}} + \sqrt [3]{719 + 24 \sqrt {687} i}}}{6}} + \left (C_{3} \sin {\left (\frac {x \sqrt {\left |{\frac {8}{3} - \frac {2 \sqrt [3]{719 + 24 \sqrt {687} i}}{3} + \frac {16 \sqrt {6}}{\sqrt {2 + \frac {97}{\sqrt [3]{719 + 24 \sqrt {687} i}} + \sqrt [3]{719 + 24 \sqrt {687} i}}} - \frac {194}{3 \sqrt [3]{719 + 24 \sqrt {687} i}}}\right |}}{2} \right )} + C_{4} \cos {\left (\frac {x \sqrt {- \frac {8}{3} + \frac {194}{3 \sqrt [3]{719 + 24 \sqrt {687} i}} - \frac {16 \sqrt {6}}{\sqrt {2 + \frac {97}{\sqrt [3]{719 + 24 \sqrt {687} i}} + \sqrt [3]{719 + 24 \sqrt {687} i}}} + \frac {2 \sqrt [3]{719 + 24 \sqrt {687} i}}{3}}}{2} \right )}\right ) e^{- \frac {\sqrt {6} x \sqrt {2 + \frac {97}{\sqrt [3]{719 + 24 \sqrt {687} i}} + \sqrt [3]{719 + 24 \sqrt {687} i}}}{6}}
\]