20.9.19 problem Problem 19
Internal
problem
ID
[3763]
Book
:
Differential
equations
and
linear
algebra,
Stephen
W.
Goode
and
Scott
A
Annin.
Fourth
edition,
2015
Section
:
Chapter
8,
Linear
differential
equations
of
order
n.
Section
8.7,
The
Variation
of
Parameters
Method.
page
556
Problem
number
:
Problem
19
Date
solved
:
Monday, January 27, 2025 at 08:01:03 AM
CAS
classification
:
[[_3rd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y&=\frac {2 \,{\mathrm e}^{x}}{x^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 22
dsolve(diff(y(x),x$3)-3*diff(y(x),x$2)+3*diff(y(x),x)-y(x)=2*x^(-2)*exp(x),y(x), singsol=all)
\[
y \left (x \right ) = {\mathrm e}^{x} \left (-2 x \ln \left (x \right )+c_{1} +c_{2} x +c_3 \,x^{2}\right )
\]
✓ Solution by Mathematica
Time used: 0.373 (sec). Leaf size: 627
DSolve[D[y[x],{x,3}]-6*D[y[x],{x,2}]+3*D[y[x],x]-y[x]==2*x^(-2)*Exp[x],y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {2 i \left (\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]\right ) \exp \left (x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right ) \left (-\frac {\exp \left (x-x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right )}{x}-\left (\left (-1+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right ) \operatorname {ExpIntegralEi}\left (x-x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right )\right )\right )}{3 \sqrt {39}}+\frac {2 i \left (\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right ) \exp \left (x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,1\right ]\right ) \left (\left (\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]-5\right ) \operatorname {ExpIntegralEi}\left (x \left (-5+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right )\right )-\frac {\exp \left (x \left (\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]-5\right )\right )}{x}\right )}{3 \sqrt {39}}-\frac {2 i \left (\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right ) \exp \left (x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]\right ) \left (-\frac {\exp \left (x-x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]\right )}{x}-\left (\left (-1+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]\right ) \operatorname {ExpIntegralEi}\left (x-x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]\right )\right )\right )}{3 \sqrt {39}}+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,2\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,3\right ]\right )+c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+3 \text {$\#$1}-1\&,1\right ]\right )
\]