50.14.10 problem 2(b)
Internal
problem
ID
[8048]
Book
:
Differential
Equations:
Theory,
Technique,
and
Practice
by
George
Simmons,
Steven
Krantz.
McGraw-Hill
NY.
2007.
1st
Edition.
Section
:
Chapter
2.
Problems
for
Review
and
Discovery.
Drill
excercises.
Page
105
Problem
number
:
2(b)
Date
solved
:
Sunday, March 30, 2025 at 12:41:22 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }-y^{\prime }+4 y&=x \end{align*}
With initial conditions
\begin{align*} y \left (1\right )&=2\\ y^{\prime }\left (1\right )&=1 \end{align*}
✓ Maple. Time used: 0.192 (sec). Leaf size: 73
ode:=diff(diff(y(x),x),x)-diff(y(x),x)+4*y(x) = x;
ic:=y(1) = 2, D(y)(1) = 1;
dsolve([ode,ic],y(x), singsol=all);
\[
y = \frac {\left (\left (\sqrt {15}\, \sin \left (\frac {\sqrt {15}}{2}\right )+135 \cos \left (\frac {\sqrt {15}}{2}\right )\right ) \cos \left (\frac {\sqrt {15}\, x}{2}\right )-\sin \left (\frac {\sqrt {15}\, x}{2}\right ) \left (\sqrt {15}\, \cos \left (\frac {\sqrt {15}}{2}\right )-135 \sin \left (\frac {\sqrt {15}}{2}\right )\right )\right ) {\mathrm e}^{-\frac {1}{2}+\frac {x}{2}}}{80}+\frac {x}{4}+\frac {1}{16}
\]
✓ Mathematica. Time used: 0.034 (sec). Leaf size: 67
ode=D[y[x],{x,2}]-D[y[x],x]+4*y[x]==x;
ic={y[1]==2,Derivative[1][y][1]==1};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {1}{80} \left (20 x-\sqrt {15} e^{\frac {x-1}{2}} \sin \left (\frac {1}{2} \sqrt {15} (x-1)\right )+135 e^{\frac {x-1}{2}} \cos \left (\frac {1}{2} \sqrt {15} (x-1)\right )+5\right )
\]
✓ Sympy. Time used: 0.289 (sec). Leaf size: 218
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x + 4*y(x) - Derivative(y(x), x) + Derivative(y(x), (x, 2)),0)
ics = {y(1): 2, Subs(Derivative(y(x), x), x, 1): 1}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {x}{4} + \left (\left (- \frac {\sqrt {15} \cos {\left (\frac {\sqrt {15}}{2} \right )}}{80 e^{\frac {1}{2}} \cos ^{2}{\left (\frac {\sqrt {15}}{2} \right )} + 80 e^{\frac {1}{2}} \sin ^{2}{\left (\frac {\sqrt {15}}{2} \right )}} + \frac {135 \sin {\left (\frac {\sqrt {15}}{2} \right )}}{80 e^{\frac {1}{2}} \cos ^{2}{\left (\frac {\sqrt {15}}{2} \right )} + 80 e^{\frac {1}{2}} \sin ^{2}{\left (\frac {\sqrt {15}}{2} \right )}}\right ) \sin {\left (\frac {\sqrt {15} x}{2} \right )} + \left (\frac {135 \cos {\left (\frac {\sqrt {15}}{2} \right )}}{80 e^{\frac {1}{2}} \cos ^{2}{\left (\frac {\sqrt {15}}{2} \right )} + 80 e^{\frac {1}{2}} \sin ^{2}{\left (\frac {\sqrt {15}}{2} \right )}} + \frac {\sqrt {15} \sin {\left (\frac {\sqrt {15}}{2} \right )}}{80 e^{\frac {1}{2}} \cos ^{2}{\left (\frac {\sqrt {15}}{2} \right )} + 80 e^{\frac {1}{2}} \sin ^{2}{\left (\frac {\sqrt {15}}{2} \right )}}\right ) \cos {\left (\frac {\sqrt {15} x}{2} \right )}\right ) e^{\frac {x}{2}} + \frac {1}{16}
\]