84.21.9 problem 13.9

Internal problem ID [22235]
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. Solved problems. Page 67
Problem number : 13.9
Date solved : Thursday, October 02, 2025 at 08:36:30 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\left (6\right )}-5 y^{\prime \prime \prime \prime }+16 y^{\prime \prime \prime }-16 y^{\prime }-32 y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 37

ode:=diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x)-5*diff(diff(diff(diff(y(x),x),x),x),x)+16*diff(diff(diff(y(x),x),x),x)-16*diff(y(x),x)-32*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \moverset {6}{\munderset {\textit {\_a} =1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-5 \textit {\_Z}^{4}+16 \textit {\_Z}^{3}-16 \textit {\_Z} -32, \operatorname {index} =\textit {\_a} \right ) x} \textit {\_C}_{\textit {\_a}} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 204

ode=D[y[x],{x,6}]-5*D[y[x],{x,4}]+16*D[y[x],{x,3}]-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_3 \exp \left (x \text {Root}\left [\text {$\#$1}^6-5 \text {$\#$1}^4+16 \text {$\#$1}^3-16 \text {$\#$1}-32\&,3\right ]\right )+c_4 \exp \left (x \text {Root}\left [\text {$\#$1}^6-5 \text {$\#$1}^4+16 \text {$\#$1}^3-16 \text {$\#$1}-32\&,4\right ]\right )+c_5 \exp \left (x \text {Root}\left [\text {$\#$1}^6-5 \text {$\#$1}^4+16 \text {$\#$1}^3-16 \text {$\#$1}-32\&,5\right ]\right )+c_6 \exp \left (x \text {Root}\left [\text {$\#$1}^6-5 \text {$\#$1}^4+16 \text {$\#$1}^3-16 \text {$\#$1}-32\&,6\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^6-5 \text {$\#$1}^4+16 \text {$\#$1}^3-16 \text {$\#$1}-32\&,2\right ]\right )+c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^6-5 \text {$\#$1}^4+16 \text {$\#$1}^3-16 \text {$\#$1}-32\&,1\right ]\right ) \end{align*}

✓ Sympy. Time used: 0.776 (sec). Leaf size: 199

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-32*y(x) - 16*Derivative(y(x), x) + 16*Derivative(y(x), (x, 3)) - 5*Derivative(y(x), (x, 4)) + Derivative(y(x), (x, 6)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{5} e^{x \operatorname {CRootOf} {\left (x^{6} - 5 x^{4} + 16 x^{3} - 16 x - 32, 0\right )}} + C_{6} e^{x \operatorname {CRootOf} {\left (x^{6} - 5 x^{4} + 16 x^{3} - 16 x - 32, 1\right )}} + \left (C_{1} \sin {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{6} - 5 x^{4} + 16 x^{3} - 16 x - 32, 2\right )}\right )} \right )} + C_{2} \cos {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{6} - 5 x^{4} + 16 x^{3} - 16 x - 32, 2\right )}\right )} \right )}\right ) e^{x \operatorname {re}{\left (\operatorname {CRootOf} {\left (x^{6} - 5 x^{4} + 16 x^{3} - 16 x - 32, 2\right )}\right )}} + \left (C_{3} \sin {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{6} - 5 x^{4} + 16 x^{3} - 16 x - 32, 4\right )}\right )} \right )} + C_{4} \cos {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{6} - 5 x^{4} + 16 x^{3} - 16 x - 32, 4\right )}\right )} \right )}\right ) e^{x \operatorname {re}{\left (\operatorname {CRootOf} {\left (x^{6} - 5 x^{4} + 16 x^{3} - 16 x - 32, 4\right )}\right )}} \]