73.1.9 problem 2.2 (i)
Internal
problem
ID
[14916]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
2.
Integration
and
differential
equations.
Additional
exercises.
page
32
Problem
number
:
2.2
(i)
Date
solved
:
Monday, March 31, 2025 at 01:03:54 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+3 y^{\prime }+8 y&={\mathrm e}^{-x^{2}} \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 116
ode:=diff(diff(y(x),x),x)+3*diff(y(x),x)+8*y(x) = exp(-x^2);
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {{\mathrm e}^{-\frac {3 x}{2}} \left (-\sqrt {\pi }\, \operatorname {erf}\left (x -\frac {3}{4}-\frac {i \sqrt {23}}{4}\right ) \sqrt {23}\, \left (i \cos \left (\frac {\sqrt {23}\, x}{2}\right )+\sin \left (\frac {\sqrt {23}\, x}{2}\right )\right ) {\mathrm e}^{\frac {3 i \sqrt {23}}{8}-\frac {7}{8}}+\sqrt {23}\, \left (i \cos \left (\frac {\sqrt {23}\, x}{2}\right )-\sin \left (\frac {\sqrt {23}\, x}{2}\right )\right ) \sqrt {\pi }\, \operatorname {erf}\left (x -\frac {3}{4}+\frac {i \sqrt {23}}{4}\right ) {\mathrm e}^{-\frac {7}{8}-\frac {3 i \sqrt {23}}{8}}-46 \cos \left (\frac {\sqrt {23}\, x}{2}\right ) c_1 -46 \sin \left (\frac {\sqrt {23}\, x}{2}\right ) c_2 \right )}{46}
\]
✓ Mathematica. Time used: 1.544 (sec). Leaf size: 146
ode=D[y[x],{x,2}]+3*D[y[x],x]+8*y[x]==Exp[-x^2];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to e^{-3 x/2} \left (\cos \left (\frac {\sqrt {23} x}{2}\right ) \int _1^x-\frac {2 e^{\frac {1}{2} (3-2 K[2]) K[2]} \sin \left (\frac {1}{2} \sqrt {23} K[2]\right )}{\sqrt {23}}dK[2]+\sin \left (\frac {\sqrt {23} x}{2}\right ) \int _1^x\frac {2 e^{\frac {1}{2} (3-2 K[1]) K[1]} \cos \left (\frac {1}{2} \sqrt {23} K[1]\right )}{\sqrt {23}}dK[1]+c_2 \cos \left (\frac {\sqrt {23} x}{2}\right )+c_1 \sin \left (\frac {\sqrt {23} x}{2}\right )\right )
\]
✓ Sympy. Time used: 36.639 (sec). Leaf size: 97
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(8*y(x) + 3*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - exp(-x**2),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (\left (C_{1} - \frac {2 \sqrt {23} \int e^{\frac {3 x}{2}} e^{- x^{2}} \sin {\left (\frac {\sqrt {23} x}{2} \right )}\, dx}{23}\right ) \cos {\left (\frac {\sqrt {23} x}{2} \right )} + \left (C_{2} + \frac {2 \sqrt {23} \int e^{\frac {3 x}{2}} e^{- x^{2}} \cos {\left (\frac {\sqrt {23} x}{2} \right )}\, dx}{23}\right ) \sin {\left (\frac {\sqrt {23} x}{2} \right )}\right ) e^{- \frac {3 x}{2}}
\]