7.11.19 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
:
Wednesday, February 05, 2025 at 03:23:48 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*}
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 235
dsolve(diff(y(x),x$5)+2*diff(y(x),x$3)+2*diff(y(x),x$2)=3*x^2-1,y(x), singsol=all)
\[
y = \frac {\left (\int \left (\int \left (2 \cos \left (\frac {\left (27+3 \sqrt {3}\, \sqrt {35}\right )^{{1}/{3}} \sqrt {3}\, \left (\sqrt {3}\, \sqrt {35}\, \left (27+3 \sqrt {3}\, \sqrt {35}\right )^{{1}/{3}}-9 \left (27+3 \sqrt {3}\, \sqrt {35}\right )^{{1}/{3}}+12\right ) x}{72}\right ) {\mathrm e}^{-\frac {\left (27+3 \sqrt {105}\right )^{{1}/{3}} x \left (-12+\left (-9+\sqrt {105}\right ) \left (27+3 \sqrt {105}\right )^{{1}/{3}}\right )}{72}} c_2 +2 \sin \left (\frac {\left (27+3 \sqrt {3}\, \sqrt {35}\right )^{{1}/{3}} \sqrt {3}\, \left (\sqrt {3}\, \sqrt {35}\, \left (27+3 \sqrt {3}\, \sqrt {35}\right )^{{1}/{3}}-9 \left (27+3 \sqrt {3}\, \sqrt {35}\right )^{{1}/{3}}+12\right ) x}{72}\right ) {\mathrm e}^{-\frac {\left (27+3 \sqrt {105}\right )^{{1}/{3}} x \left (-12+\left (-9+\sqrt {105}\right ) \left (27+3 \sqrt {105}\right )^{{1}/{3}}\right )}{72}} c_3 +2 c_1 \,{\mathrm e}^{\frac {\left (27+3 \sqrt {105}\right )^{{1}/{3}} x \left (-12+\left (-9+\sqrt {105}\right ) \left (27+3 \sqrt {105}\right )^{{1}/{3}}\right )}{36}}+3 x^{2}-6 x \right )d x +5 x \right )d x \right )}{2}+c_4 x +c_5
\]
✓ Solution by Mathematica
Time used: 0.196 (sec). Leaf size: 150
DSolve[D[y[x],{x,5}]+2*D[y[x],{x,3}]+2*D[y[x],{x,2}]==3*x^2-1,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {1}{2} \left (\frac {2 c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+2\&,3\right ]\right )}{\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+2\&,3\right ]^2}+\frac {2 c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+2\&,2\right ]\right )}{\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+2\&,2\right ]^2}+\frac {2 c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+2\&,1\right ]\right )}{\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+2\&,1\right ]^2}+\frac {x^4}{4}-x^3+\frac {5 x^2}{2}\right )+c_5 x+c_4
\]