89.13.13 problem 13
Internal
problem
ID
[24596]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
8.
Linear
Differential
Equations
with
constant
coefficients.
Exercises
at
page
127
Problem
number
:
13
Date
solved
:
Thursday, October 02, 2025 at 10:46:24 PM
CAS
classification
:
[[_high_order, _missing_x]]
\begin{align*} y^{\left (5\right )}+y^{\prime \prime \prime \prime }-7 y^{\prime \prime \prime }-11 y^{\prime \prime }-8 y^{\prime }+12 y&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 40
ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)+diff(diff(diff(diff(y(x),x),x),x),x)-7*diff(diff(diff(y(x),x),x),x)-11*diff(diff(y(x),x),x)-8*diff(y(x),x)+12*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \moverset {5}{\munderset {\textit {\_a} =1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{5}+\textit {\_Z}^{4}-7 \textit {\_Z}^{3}-11 \textit {\_Z}^{2}-8 \textit {\_Z} +12, \operatorname {index} =\textit {\_a} \right ) x} \textit {\_C}_{\textit {\_a}}
\]
✓ Mathematica. Time used: 0.002 (sec). Leaf size: 191
ode=D[y[x],{x,5}]+D[y[x],{x,4}]-7*D[y[x],{x,3}]-11*D[y[x],{x,2}]-8*D[y[x],{x,1}]+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 {$\#$1}^5+\text {$\#$1}^4-7 \text {$\#$1}^3-11 \text {$\#$1}^2-8 \text {$\#$1}+12\&,2\right ]\right )+c_4 \exp \left (x \text {Root}\left [\text {$\#$1}^5+\text {$\#$1}^4-7 \text {$\#$1}^3-11 \text {$\#$1}^2-8 \text {$\#$1}+12\&,4\right ]\right )+c_5 \exp \left (x \text {Root}\left [\text {$\#$1}^5+\text {$\#$1}^4-7 \text {$\#$1}^3-11 \text {$\#$1}^2-8 \text {$\#$1}+12\&,5\right ]\right )+c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^5+\text {$\#$1}^4-7 \text {$\#$1}^3-11 \text {$\#$1}^2-8 \text {$\#$1}+12\&,1\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^5+\text {$\#$1}^4-7 \text {$\#$1}^3-11 \text {$\#$1}^2-8 \text {$\#$1}+12\&,3\right ]\right ) \end{align*}
✓ Sympy. Time used: 0.409 (sec). Leaf size: 168
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(12*y(x) - 8*Derivative(y(x), x) - 11*Derivative(y(x), (x, 2)) - 7*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)) + Derivative(y(x), (x, 5)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{3} e^{x \operatorname {CRootOf} {\left (x^{5} + x^{4} - 7 x^{3} - 11 x^{2} - 8 x + 12, 0\right )}} + C_{4} e^{x \operatorname {CRootOf} {\left (x^{5} + x^{4} - 7 x^{3} - 11 x^{2} - 8 x + 12, 1\right )}} + C_{5} e^{x \operatorname {CRootOf} {\left (x^{5} + x^{4} - 7 x^{3} - 11 x^{2} - 8 x + 12, 2\right )}} + \left (C_{1} \sin {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} + x^{4} - 7 x^{3} - 11 x^{2} - 8 x + 12, 3\right )}\right )} \right )} + C_{2} \cos {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} + x^{4} - 7 x^{3} - 11 x^{2} - 8 x + 12, 3\right )}\right )} \right )}\right ) e^{x \operatorname {re}{\left (\operatorname {CRootOf} {\left (x^{5} + x^{4} - 7 x^{3} - 11 x^{2} - 8 x + 12, 3\right )}\right )}}
\]