Internal problem ID [4467]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_linear, `class A`]]
\[ \boxed {x^{\prime }-\alpha +\beta \cos \left (\frac {\pi t}{12}\right )+k x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = x_{0}] \end {align*}
✓ Solution by Maple
Time used: 0.047 (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)
\[ x \left (t \right ) = \frac {-144 \cos \left (\frac {\pi t}{12}\right ) \beta \,k^{2}-12 \pi \sin \left (\frac {\pi t}{12}\right ) \beta k +\left (144 k^{3} x_{0} +144 \left (\beta -\alpha \right ) k^{2}+\pi ^{2} k x_{0} -\pi ^{2} \alpha \right ) {\mathrm e}^{-k t}+144 \alpha \,k^{2}+\pi ^{2} \alpha }{\pi ^{2} k +144 k^{3}} \]
✓ Solution by Mathematica
Time used: 0.251 (sec). Leaf size: 54
DSolve[{x'[t]==\[Alpha]-\[Beta]*Cos[Pi*t/12]-k*x[t],{}},x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -\frac {12 \beta \left (12 k \cos \left (\frac {\pi t}{12}\right )+\pi \sin \left (\frac {\pi t}{12}\right )\right )}{144 k^2+\pi ^2}+\frac {\alpha }{k}+c_1 e^{-k t} \\ \end{align*}