71.9.2 problem 2
Internal
problem
ID
[14410]
Book
:
Ordinary
Differential
Equations
by
Charles
E.
Roberts,
Jr.
CRC
Press.
2010
Section
:
Chapter
4.
N-th
Order
Linear
Differential
Equations.
Exercises
4.1,
page
186
Problem
number
:
2
Date
solved
:
Monday, March 31, 2025 at 12:26:20 PM
CAS
classification
:
[[_3rd_order, _missing_y]]
\begin{align*} x y^{\prime \prime \prime }+x y^{\prime }&=4 \end{align*}
With initial conditions
\begin{align*} y \left (1\right )&=0\\ y^{\prime }\left (1\right )&=1\\ y^{\prime \prime }\left (1\right )&=-1 \end{align*}
✓ Maple. Time used: 0.095 (sec). Leaf size: 44
ode:=x*diff(diff(diff(y(x),x),x),x)+x*diff(y(x),x) = 4;
ic:=y(1) = 0, D(y)(1) = 1, (D@@2)(y)(1) = -1;
dsolve([ode,ic],y(x), singsol=all);
\[
y = \left (4 \,\operatorname {Ci}\left (1\right )-4 \,\operatorname {Ci}\left (x \right )+\cos \left (1\right )-\sin \left (1\right )\right ) \cos \left (x \right )+\left (4 \,\operatorname {Si}\left (1\right )-4 \,\operatorname {Si}\left (x \right )+\cos \left (1\right )+\sin \left (1\right )\right ) \sin \left (x \right )+4 \ln \left (x \right )-1
\]
✓ Mathematica. Time used: 60.029 (sec). Leaf size: 63
ode=x*D[y[x],{x,3}]+x*D[y[x],x]==4;
ic={y[1]==0,Derivative[1][y][1]==1,Derivative[2][y][1]==-1};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \int _1^x\left (\cos (1-K[2])+\sin (1-K[2])+4 \operatorname {CosIntegral}(K[2]) \sin (K[2])-4 \operatorname {CosIntegral}(1) \sin (K[2])+\cos (K[2]) \int _1^{K[2]}-\frac {4 \sin (K[1])}{K[1]}dK[1]\right )dK[2]
\]
✓ Sympy. Time used: 0.863 (sec). Leaf size: 201
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*Derivative(y(x), x) + x*Derivative(y(x), (x, 3)) - 4,0)
ics = {y(1): 0, Subs(Derivative(y(x), x), x, 1): 1, Subs(Derivative(y(x), (x, 2)), x, 1): -1}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (- 4 \operatorname {Si}{\left (x \right )} + \frac {- 3 \sin {\left (1 \right )} - 3 \cos {\left (1 \right )} + 4 \cos ^{3}{\left (1 \right )} + 4 \sin {\left (1 \right )} \cos ^{2}{\left (1 \right )} + 4 \cos ^{2}{\left (1 \right )} \operatorname {Si}{\left (1 \right )} + 4 \sin ^{2}{\left (1 \right )} \cos {\left (1 \right )} + 4 \sin ^{3}{\left (1 \right )} + 4 \sin ^{2}{\left (1 \right )} \operatorname {Si}{\left (1 \right )}}{\cos ^{2}{\left (1 \right )} + \sin ^{2}{\left (1 \right )}}\right ) \sin {\left (x \right )} + \left (4 \log {\left (x \right )} - 2 \log {\left (x^{2} \right )} - 4 \operatorname {Ci}{\left (x \right )} + \frac {- 4 \sin ^{3}{\left (1 \right )} - 3 \cos {\left (1 \right )} - 4 \sin {\left (1 \right )} \cos ^{2}{\left (1 \right )} + 4 \cos ^{2}{\left (1 \right )} \operatorname {Ci}{\left (1 \right )} + 4 \cos ^{3}{\left (1 \right )} + 4 \sin ^{2}{\left (1 \right )} \operatorname {Ci}{\left (1 \right )} + 4 \sin ^{2}{\left (1 \right )} \cos {\left (1 \right )} + 3 \sin {\left (1 \right )}}{\cos ^{2}{\left (1 \right )} + \sin ^{2}{\left (1 \right )}}\right ) \cos {\left (x \right )} + 4 \log {\left (x \right )} - 4 \sin ^{2}{\left (1 \right )} - 4 \cos ^{2}{\left (1 \right )} + 3
\]