65.11.7 problem 7
Internal
problem
ID
[15803]
Book
:
Ordinary
Differential
Equations
by
Charles
E.
Roberts,
Jr.
CRC
Press.
2010
Section
:
Chapter
4.
N-th
Order
Linear
Differential
Equations.
Exercises
4.4,
page
218
Problem
number
:
7
Date
solved
:
Thursday, October 02, 2025 at 10:28:11 AM
CAS
classification
:
[[_high_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\left (6\right )}-12 y^{\left (5\right )}+63 y^{\prime \prime \prime \prime }-18 y^{\prime \prime \prime }+315 y^{\prime \prime }-300 y^{\prime }+125 y&={\mathrm e}^{x} \left (48 \cos \left (x \right )+96 \sin \left (x \right )\right ) \end{align*}
✓ Maple. Time used: 0.352 (sec). Leaf size: 4681
ode:=diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x)-12*diff(diff(diff(diff(diff(y(x),x),x),x),x),x)+63*diff(diff(diff(diff(y(x),x),x),x),x)-18*diff(diff(diff(y(x),x),x),x)+315*diff(diff(y(x),x),x)-300*diff(y(x),x)+125*y(x) = exp(x)*(48*cos(x)+96*sin(x));
dsolve(ode,y(x), singsol=all);
\[
\text {Expression too large to display}
\]
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 292
ode=D[y[x],{x,6}]-12*D[y[x],{x,5}]+63*D[y[x],{x,4}]-18*D[y[x],{x,3}]+315*D[y[x],{x,2}]-300*D[y[x],x]+125*y[x]==Exp[x]*(48*Cos[x]+96*Sin[x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^6-12 \text {$\#$1}^5+63 \text {$\#$1}^4-18 \text {$\#$1}^3+315 \text {$\#$1}^2-300 \text {$\#$1}+125\&,3\right ]\right )+c_4 \exp \left (x \text {Root}\left [\text {$\#$1}^6-12 \text {$\#$1}^5+63 \text {$\#$1}^4-18 \text {$\#$1}^3+315 \text {$\#$1}^2-300 \text {$\#$1}+125\&,4\right ]\right )+c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^6-12 \text {$\#$1}^5+63 \text {$\#$1}^4-18 \text {$\#$1}^3+315 \text {$\#$1}^2-300 \text {$\#$1}+125\&,1\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^6-12 \text {$\#$1}^5+63 \text {$\#$1}^4-18 \text {$\#$1}^3+315 \text {$\#$1}^2-300 \text {$\#$1}+125\&,2\right ]\right )+c_5 \exp \left (x \text {Root}\left [\text {$\#$1}^6-12 \text {$\#$1}^5+63 \text {$\#$1}^4-18 \text {$\#$1}^3+315 \text {$\#$1}^2-300 \text {$\#$1}+125\&,5\right ]\right )+c_6 \exp \left (x \text {Root}\left [\text {$\#$1}^6-12 \text {$\#$1}^5+63 \text {$\#$1}^4-18 \text {$\#$1}^3+315 \text {$\#$1}^2-300 \text {$\#$1}+125\&,6\right ]\right )-\frac {48 e^x (352 \sin (x)+1011 \cos (x))}{229205} \end{align*}
✓ Sympy. Time used: 1.326 (sec). Leaf size: 338
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((-96*sin(x) - 48*cos(x))*exp(x) + 125*y(x) - 300*Derivative(y(x), x) + 315*Derivative(y(x), (x, 2)) - 18*Derivative(y(x), (x, 3)) + 63*Derivative(y(x), (x, 4)) - 12*Derivative(y(x), (x, 5)) + Derivative(y(x), (x, 6)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (C_{1} \sin {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{6} - 12 x^{5} + 63 x^{4} - 18 x^{3} + 315 x^{2} - 300 x + 125, 0\right )}\right )} \right )} + C_{2} \cos {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{6} - 12 x^{5} + 63 x^{4} - 18 x^{3} + 315 x^{2} - 300 x + 125, 0\right )}\right )} \right )}\right ) e^{x \operatorname {re}{\left (\operatorname {CRootOf} {\left (x^{6} - 12 x^{5} + 63 x^{4} - 18 x^{3} + 315 x^{2} - 300 x + 125, 0\right )}\right )}} + \left (C_{3} \sin {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{6} - 12 x^{5} + 63 x^{4} - 18 x^{3} + 315 x^{2} - 300 x + 125, 2\right )}\right )} \right )} + C_{4} \cos {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{6} - 12 x^{5} + 63 x^{4} - 18 x^{3} + 315 x^{2} - 300 x + 125, 2\right )}\right )} \right )}\right ) e^{x \operatorname {re}{\left (\operatorname {CRootOf} {\left (x^{6} - 12 x^{5} + 63 x^{4} - 18 x^{3} + 315 x^{2} - 300 x + 125, 2\right )}\right )}} + \left (C_{5} \sin {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{6} - 12 x^{5} + 63 x^{4} - 18 x^{3} + 315 x^{2} - 300 x + 125, 4\right )}\right )} \right )} + C_{6} \cos {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{6} - 12 x^{5} + 63 x^{4} - 18 x^{3} + 315 x^{2} - 300 x + 125, 4\right )}\right )} \right )}\right ) e^{x \operatorname {re}{\left (\operatorname {CRootOf} {\left (x^{6} - 12 x^{5} + 63 x^{4} - 18 x^{3} + 315 x^{2} - 300 x + 125, 4\right )}\right )}} - \frac {16896 e^{x} \sin {\left (x \right )}}{229205} - \frac {48528 e^{x} \cos {\left (x \right )}}{229205}
\]