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 : Monday, January 27, 2025 at 01:56:27 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*}

✓ Solution by Maple

Time used: 0.045 (sec). Leaf size: 86

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

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

✓ Solution by Mathematica

Time used: 0.269 (sec). Leaf size: 64

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

\[ x(t)\to -\frac {12 \pi \beta \sin \left (\frac {\pi t}{12}\right )}{144 k^2+\pi ^2}-\frac {144 \beta k \cos \left (\frac {\pi t}{12}\right )}{144 k^2+\pi ^2}+\frac {\alpha }{k}+c_1 e^{-k t} \]