74.1.3 problem 4
Internal
problem
ID
[15783]
Book
:
INTRODUCTORY
DIFFERENTIAL
EQUATIONS.
Martha
L.
Abell,
James
P.
Braselton.
Fourth
edition
2014.
ElScAe.
2014
Section
:
Chapter
1.
Introduction
to
Differential
Equations.
Exercises
1.1,
page
10
Problem
number
:
4
Date
solved
:
Tuesday, January 28, 2025 at 08:07:09 AM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime \prime }-2 y^{\prime \prime }+5 y^{\prime }+y&={\mathrm e}^{x} \end{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 251
dsolve(diff(y(x),x$3)-2*diff(y(x),x$2)+5*diff(y(x),x)+y(x)=exp(x),y(x), singsol=all)
\[
y = \frac {{\mathrm e}^{x}}{5}+c_{1} {\mathrm e}^{\frac {\left (15 \sqrt {69}\, \left (404+60 \sqrt {69}\right )^{{2}/{3}}-101 \left (404+60 \sqrt {69}\right )^{{2}/{3}}-484 \left (404+60 \sqrt {69}\right )^{{1}/{3}}+1936\right ) x}{2904}}+c_{2} {\mathrm e}^{-\frac {\left (15 \sqrt {69}\, \left (404+60 \sqrt {69}\right )^{{2}/{3}}-101 \left (404+60 \sqrt {69}\right )^{{2}/{3}}-484 \left (404+60 \sqrt {69}\right )^{{1}/{3}}-3872\right ) x}{5808}} \cos \left (\frac {\left (404+60 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}} \sqrt {3}\, \left (15 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {23}-101 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}}+484\right ) x}{5808}\right )+c_{3} {\mathrm e}^{-\frac {\left (15 \sqrt {69}\, \left (404+60 \sqrt {69}\right )^{{2}/{3}}-101 \left (404+60 \sqrt {69}\right )^{{2}/{3}}-484 \left (404+60 \sqrt {69}\right )^{{1}/{3}}-3872\right ) x}{5808}} \sin \left (\frac {\left (404+60 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}} \sqrt {3}\, \left (15 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {23}-101 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}}+484\right ) x}{5808}\right )
\]
✓ Solution by Mathematica
Time used: 0.358 (sec). Leaf size: 504
DSolve[D[y[x],{x,3}]-2*D[y[x],{x,2}]+5*D[y[x],x]+y[x]==Exp[x],y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \exp \left (x \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,3\right ]\right ) \int _1^x\frac {i \exp \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,2\right ] K[3]+\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,1\right ] K[3]-K[3]\right ) \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,2\right ]\right )}{5 \sqrt {23}}dK[3]+\exp \left (x \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,2\right ]\right ) \int _1^x-\frac {i \exp \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,3\right ] K[2]+\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,1\right ] K[2]-K[2]\right ) \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,3\right ]\right )}{5 \sqrt {23}}dK[2]+\exp \left (x \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,1\right ]\right ) \int _1^x\frac {i \exp \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,3\right ] K[1]+\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,2\right ] K[1]-K[1]\right ) \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,3\right ]\right )}{5 \sqrt {23}}dK[1]+c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,1\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,2\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,3\right ]\right )
\]