36.2.27 problem 37

Internal problem ID [6320]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.3, Linear equations. Exercises. page 54
Problem number : 37
Date solved : Sunday, March 30, 2025 at 10:51:26 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} x^{\prime }&=\alpha -\beta \cos \left (\frac {\pi t}{12}\right )-k x \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=x_{0} \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 86

ode:=diff(x(t),t) = alpha-beta*cos(1/12*Pi*t)-k*x(t); 
ic:=x(0) = x__0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x = \frac {-144 \cos \left (\frac {\pi t}{12}\right ) \beta \,k^{2}-12 \sin \left (\frac {\pi t}{12}\right ) \pi \beta k +\left (144 k^{3} x_{0} +144 \left (\beta -\alpha \right ) k^{2}+\pi ^{2} k x_{0} -\pi ^{2} \alpha \right ) {\mathrm e}^{-k t}+144 \alpha \,k^{2}+\pi ^{2} \alpha }{\pi ^{2} k +144 k^{3}} \]

✓ Mathematica. Time used: 0.3 (sec). Leaf size: 113

ode=D[x[t],t]==\[Alpha]-\[Beta]*Cos[Pi*t/12]-k*x[t]; 
ic={x[0]==x0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {e^{-k t} \left (-\pi ^2 \alpha +144 k^3 \text {x0}-144 \alpha k^2+144 \beta k^2+144 \alpha k^2 e^{k t}-144 \beta k^2 e^{k t} \cos \left (\frac {\pi t}{12}\right )+\pi ^2 \alpha e^{k t}-12 \pi \beta k e^{k t} \sin \left (\frac {\pi t}{12}\right )+\pi ^2 k \text {x0}\right )}{k \left (144 k^2+\pi ^2\right )} \]

✓ Sympy. Time used: 0.218 (sec). Leaf size: 94

from sympy import * 
t = symbols("t") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
k = symbols("k") 
x = Function("x") 
ode = Eq(-Alpha + BETA*cos(pi*t/12) + k*x(t) + Derivative(x(t), t),0) 
ics = {x(0): x__0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {\mathrm {A}}{k} - \frac {144 \beta k \cos {\left (\frac {\pi t}{12} \right )}}{144 k^{2} + \pi ^{2}} - \frac {12 \pi \beta \sin {\left (\frac {\pi t}{12} \right )}}{144 k^{2} + \pi ^{2}} + \frac {\left (- 144 \mathrm {A} k^{2} - \pi ^{2} \mathrm {A} + 144 \beta k^{2} + 144 k^{3} x^{0} + \pi ^{2} k x^{0}\right ) e^{- k t}}{144 k^{3} + \pi ^{2} k} \]