87.16.10 problem 10
Internal
problem
ID
[23579]
Book
:
Ordinary
differential
equations
with
modern
applications.
Ladas,
G.
E.
and
Finizio,
N.
Wadsworth
Publishing.
California.
1978.
ISBN
0-534-00552-7.
QA372.F56
Section
:
Chapter
2.
Linear
differential
equations.
Exercise
at
page
119
Problem
number
:
10
Date
solved
:
Thursday, October 02, 2025 at 09:43:08 PM
CAS
classification
:
[[_3rd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime \prime }-y&=3 \ln \left (x \right ) \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 121
ode:=diff(diff(diff(y(x),x),x),x)-y(x) = 3*ln(x);
dsolve(ode,y(x), singsol=all);
\[
y = -{\mathrm e}^{-\frac {x}{2}} \left (i \sin \left (\frac {\sqrt {3}\, x}{2}\right )+\cos \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \operatorname {Ei}_{1}\left (\frac {x \left (i \sqrt {3}-1\right )}{2}\right )+{\mathrm e}^{-\frac {x}{2}} \left (i \sin \left (\frac {\sqrt {3}\, x}{2}\right )-\cos \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \operatorname {Ei}_{1}\left (-\frac {x \left (1+i \sqrt {3}\right )}{2}\right )+c_2 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_3 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+c_1 \,{\mathrm e}^{x}-\operatorname {Ei}_{1}\left (x \right ) {\mathrm e}^{x}-3 \ln \left (x \right )
\]
✓ Mathematica. Time used: 0.235 (sec). Leaf size: 181
ode=D[y[x],{x,3}]-y[x]==3*Log[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to e^{-x/2} \left (e^{3 x/2} \operatorname {ExpIntegralEi}(-x)-i \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (i \sqrt {3} x+x\right )\right ) \sin \left (\frac {\sqrt {3} x}{2}\right )+\operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (i \sqrt {3} x+x\right )\right ) \cos \left (\frac {\sqrt {3} x}{2}\right )+\operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (x-i \sqrt {3} x\right )\right ) \left (\cos \left (\frac {\sqrt {3} x}{2}\right )+i \sin \left (\frac {\sqrt {3} x}{2}\right )\right )-3 e^{x/2} \log (x)+c_1 e^{3 x/2}+c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )+c_3 \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \end{align*}
✓ Sympy. Time used: 23.330 (sec). Leaf size: 99
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-y(x) - 3*log(x) + Derivative(y(x), (x, 3)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (C_{3} + \operatorname {Ei}{\left (- x \right )}\right ) e^{x} + \left (\left (C_{1} - 2 \int e^{\frac {x}{2}} \log {\left (x \right )} \sin {\left (\frac {3 \sqrt {3} x + 2 \pi }{6} \right )}\, dx\right ) \sin {\left (\frac {\sqrt {3} x}{2} \right )} + \left (C_{2} - 2 \int e^{\frac {x}{2}} \log {\left (x \right )} \cos {\left (\frac {3 \sqrt {3} x + 2 \pi }{6} \right )}\, dx\right ) \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} - \log {\left (x \right )}
\]