1.11.19 problem 19
Internal
problem
ID
[340]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
2.
Linear
Equations
of
Higher
Order.
Section
2.5
(Nonhomogeneous
equations
and
undetermined
coefficients).
Problems
at
page
161
Problem
number
:
19
Date
solved
:
Tuesday, September 30, 2025 at 03:57:33 AM
CAS
classification
:
[[_high_order, _missing_y]]
\begin{align*} y^{\left (5\right )}+2 y^{\prime \prime \prime }+2 y^{\prime \prime }&=3 x^{2}-1 \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 235
ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)+2*diff(diff(diff(y(x),x),x),x)+2*diff(diff(y(x),x),x) = 3*x^2-1;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\int \left (\int \left (2 \,{\mathrm e}^{-\frac {\left (27+3 \sqrt {105}\right )^{{1}/{3}} x \left (-12+\left (-9+\sqrt {105}\right ) \left (27+3 \sqrt {105}\right )^{{1}/{3}}\right )}{72}} \cos \left (\frac {\left (27+3 \sqrt {3}\, \sqrt {35}\right )^{{1}/{3}} \sqrt {3}\, \left (\left (27+3 \sqrt {3}\, \sqrt {35}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {35}-9 \left (27+3 \sqrt {3}\, \sqrt {35}\right )^{{1}/{3}}+12\right ) x}{72}\right ) c_2 +2 \,{\mathrm e}^{-\frac {\left (27+3 \sqrt {105}\right )^{{1}/{3}} x \left (-12+\left (-9+\sqrt {105}\right ) \left (27+3 \sqrt {105}\right )^{{1}/{3}}\right )}{72}} \sin \left (\frac {\left (27+3 \sqrt {3}\, \sqrt {35}\right )^{{1}/{3}} \sqrt {3}\, \left (\left (27+3 \sqrt {3}\, \sqrt {35}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {35}-9 \left (27+3 \sqrt {3}\, \sqrt {35}\right )^{{1}/{3}}+12\right ) x}{72}\right ) c_3 +2 c_1 \,{\mathrm e}^{\frac {\left (27+3 \sqrt {105}\right )^{{1}/{3}} x \left (-12+\left (-9+\sqrt {105}\right ) \left (27+3 \sqrt {105}\right )^{{1}/{3}}\right )}{36}}+3 x^{2}-6 x \right )d x +5 x \right )d x}{2}+c_4 x +c_5
\]
✓ Mathematica. Time used: 0.117 (sec). Leaf size: 150
ode=D[y[x],{x,5}]+2*D[y[x],{x,3}]+2*D[y[x],{x,2}]==3*x^2-1;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{2} \left (\frac {2 c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+2\&,3\right ]\right )}{\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+2\&,3\right ]^2}+\frac {2 c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+2\&,2\right ]\right )}{\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+2\&,2\right ]^2}+\frac {2 c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+2\&,1\right ]\right )}{\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+2\&,1\right ]^2}+\frac {x^4}{4}-x^3+\frac {5 x^2}{2}\right )+c_5 x+c_4 \end{align*}
✓ Sympy. Time used: 0.267 (sec). Leaf size: 216
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-3*x**2 + 2*Derivative(y(x), (x, 2)) + 2*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 5)) + 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} + C_{2} x + C_{3} e^{- \frac {\sqrt [3]{3} x \left (- \sqrt [3]{9 + \sqrt {105}} + \frac {2 \sqrt [3]{3}}{\sqrt [3]{9 + \sqrt {105}}}\right )}{6}} \sin {\left (\sqrt [6]{3} x \left (\frac {1}{\sqrt [3]{9 + \sqrt {105}}} + \frac {3^{\frac {2}{3}} \sqrt [3]{9 + \sqrt {105}}}{6}\right ) \right )} + C_{4} e^{- \frac {\sqrt [3]{3} x \left (- \sqrt [3]{9 + \sqrt {105}} + \frac {2 \sqrt [3]{3}}{\sqrt [3]{9 + \sqrt {105}}}\right )}{6}} \cos {\left (\sqrt [6]{3} x \left (\frac {1}{\sqrt [3]{9 + \sqrt {105}}} + \frac {3^{\frac {2}{3}} \sqrt [3]{9 + \sqrt {105}}}{6}\right ) \right )} + C_{5} e^{\frac {\sqrt [3]{3} x \left (- \sqrt [3]{9 + \sqrt {105}} + \frac {2 \sqrt [3]{3}}{\sqrt [3]{9 + \sqrt {105}}}\right )}{3}} + \frac {x^{4}}{8} - \frac {x^{3}}{2} + \frac {5 x^{2}}{4}
\]