30.2.27 problem 37

Internal problem ID [7455]
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 : Tuesday, September 30, 2025 at 04:36:18 PM
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,op(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: 0.142 (sec). Leaf size: 41

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]
 

\begin{align*} x(t)&\to e^{-k t} \left (\int _0^te^{k K[1]} \left (\alpha -\beta \cos \left (\frac {1}{12} \pi K[1]\right )\right )dK[1]+\text {x0}\right ) \end{align*}

✓ Sympy. Time used: 0.131 (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)
 

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