71.9.1 problem 1
Internal
problem
ID
[14409]
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
:
1
Date
solved
:
Monday, March 31, 2025 at 12:26:18 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 3 y^{\prime \prime }-2 y^{\prime }+4 y&=x \end{align*}
With initial conditions
\begin{align*} y \left (-1\right )&=2\\ y^{\prime }\left (-1\right )&=3 \end{align*}
✓ Maple. Time used: 0.201 (sec). Leaf size: 74
ode:=3*diff(diff(y(x),x),x)-2*diff(y(x),x)+4*y(x) = x;
ic:=y(-1) = 2, D(y)(-1) = 3;
dsolve([ode,ic],y(x), singsol=all);
\[
y = \frac {\left (\left (49 \sin \left (\frac {\sqrt {11}}{3}\right ) \sqrt {11}+187 \cos \left (\frac {\sqrt {11}}{3}\right )\right ) \cos \left (\frac {\sqrt {11}\, x}{3}\right )+49 \left (\cos \left (\frac {\sqrt {11}}{3}\right ) \sqrt {11}-\frac {187 \sin \left (\frac {\sqrt {11}}{3}\right )}{49}\right ) \sin \left (\frac {\sqrt {11}\, x}{3}\right )\right ) {\mathrm e}^{\frac {1}{3}+\frac {x}{3}}}{88}+\frac {x}{4}+\frac {1}{8}
\]
✓ Mathematica. Time used: 0.036 (sec). Leaf size: 67
ode=3*D[y[x],{x,2}]-2*D[y[x],x]+4*y[x]==x;
ic={y[-1]==2,Derivative[1][y][-1]==3};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {1}{88} \left (22 x+49 \sqrt {11} e^{\frac {x+1}{3}} \sin \left (\frac {1}{3} \sqrt {11} (x+1)\right )+187 e^{\frac {x+1}{3}} \cos \left (\frac {1}{3} \sqrt {11} (x+1)\right )+11\right )
\]
✓ Sympy. Time used: 0.321 (sec). Leaf size: 201
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x + 4*y(x) - 2*Derivative(y(x), x) + 3*Derivative(y(x), (x, 2)),0)
ics = {y(-1): 2, Subs(Derivative(y(x), x), x, -1): 3}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {x}{4} + \left (\left (- \frac {187 e^{\frac {1}{3}} \sin {\left (\frac {\sqrt {11}}{3} \right )}}{88 \cos ^{2}{\left (\frac {\sqrt {11}}{3} \right )} + 88 \sin ^{2}{\left (\frac {\sqrt {11}}{3} \right )}} + \frac {49 \sqrt {11} e^{\frac {1}{3}} \cos {\left (\frac {\sqrt {11}}{3} \right )}}{88 \cos ^{2}{\left (\frac {\sqrt {11}}{3} \right )} + 88 \sin ^{2}{\left (\frac {\sqrt {11}}{3} \right )}}\right ) \sin {\left (\frac {\sqrt {11} x}{3} \right )} + \left (\frac {187 e^{\frac {1}{3}} \cos {\left (\frac {\sqrt {11}}{3} \right )}}{88 \cos ^{2}{\left (\frac {\sqrt {11}}{3} \right )} + 88 \sin ^{2}{\left (\frac {\sqrt {11}}{3} \right )}} + \frac {49 \sqrt {11} e^{\frac {1}{3}} \sin {\left (\frac {\sqrt {11}}{3} \right )}}{88 \cos ^{2}{\left (\frac {\sqrt {11}}{3} \right )} + 88 \sin ^{2}{\left (\frac {\sqrt {11}}{3} \right )}}\right ) \cos {\left (\frac {\sqrt {11} x}{3} \right )}\right ) e^{\frac {x}{3}} + \frac {1}{8}
\]