88.17.1 problem 5
Internal
problem
ID
[24112]
Book
:
Elementary
Differential
Equations.
By
Lee
Roy
Wilcox
and
Herbert
J.
Curtis.
1961
first
edition.
International
texbook
company.
Scranton,
Penn.
USA.
CAT
number
61-15976
Section
:
Chapter
5.
Special
Techniques
for
Linear
Equations.
Exercises
at
page
127
Problem
number
:
5
Date
solved
:
Thursday, October 02, 2025 at 09:59:20 PM
CAS
classification
:
[[_high_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\left (5\right )}-y^{\prime \prime \prime \prime }+2 y^{\prime }-y&=x^{4}-2 x +1 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 134
ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-diff(diff(diff(diff(y(x),x),x),x),x)+2*diff(y(x),x)-y(x) = x^4-2*x+1;
dsolve(ode,y(x), singsol=all);
\[
y = -x^{4}-8 x^{3}-48 x^{2}-190 x -357+c_1 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{5}-\textit {\_Z}^{4}+2 \textit {\_Z} -1, \operatorname {index} =1\right ) x}+c_2 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{5}-\textit {\_Z}^{4}+2 \textit {\_Z} -1, \operatorname {index} =2\right ) x}+c_3 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{5}-\textit {\_Z}^{4}+2 \textit {\_Z} -1, \operatorname {index} =3\right ) x}+c_4 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{5}-\textit {\_Z}^{4}+2 \textit {\_Z} -1, \operatorname {index} =4\right ) x}+c_5 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{5}-\textit {\_Z}^{4}+2 \textit {\_Z} -1, \operatorname {index} =5\right ) x}
\]
✓ Mathematica. Time used: 0.003 (sec). Leaf size: 160
ode=D[y[x],{x,5}]-D[y[x],{x,4}]+2*D[y[x],x]-y[x]==x^4-2*x+1;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^5-\text {$\#$1}^4+2 \text {$\#$1}-1\&,1\right ]\right )+c_4 \exp \left (x \text {Root}\left [\text {$\#$1}^5-\text {$\#$1}^4+2 \text {$\#$1}-1\&,4\right ]\right )+c_5 \exp \left (x \text {Root}\left [\text {$\#$1}^5-\text {$\#$1}^4+2 \text {$\#$1}-1\&,5\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^5-\text {$\#$1}^4+2 \text {$\#$1}-1\&,2\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^5-\text {$\#$1}^4+2 \text {$\#$1}-1\&,3\right ]\right )-x^4-8 x^3-48 x^2-190 x-357 \end{align*}
✓ Sympy. Time used: 0.532 (sec). Leaf size: 146
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**4 + 2*x - y(x) + 2*Derivative(y(x), x) - Derivative(y(x), (x, 4)) + Derivative(y(x), (x, 5)) - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{5} e^{x \operatorname {CRootOf} {\left (x^{5} - x^{4} + 2 x - 1, 0\right )}} - x^{4} - 8 x^{3} - 48 x^{2} - 190 x + \left (C_{1} \sin {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} - x^{4} + 2 x - 1, 1\right )}\right )} \right )} + C_{2} \cos {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} - x^{4} + 2 x - 1, 1\right )}\right )} \right )}\right ) e^{x \operatorname {re}{\left (\operatorname {CRootOf} {\left (x^{5} - x^{4} + 2 x - 1, 1\right )}\right )}} + \left (C_{3} \sin {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} - x^{4} + 2 x - 1, 3\right )}\right )} \right )} + C_{4} \cos {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} - x^{4} + 2 x - 1, 3\right )}\right )} \right )}\right ) e^{x \operatorname {re}{\left (\operatorname {CRootOf} {\left (x^{5} - x^{4} + 2 x - 1, 3\right )}\right )}} - 357
\]