Internal problem ID [10453]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY.
2015.
Section: Chapter 2, Second order linear equations. Section 2.3.1 Nonhomogeneous Equations:
Undetermined Coefficients. Exercises page 110
Problem number: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }-b x^{\prime }+x-\sin \left (2 t \right )=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 135
dsolve([diff(x(t),t$2)-b*diff(x(t),t)+x(t)=sin(2*t),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
\[ x \left (t \right ) = \frac {\left (-\sqrt {b^{2}-4}\, b^{2}-b^{3}-6 \sqrt {b^{2}-4}+4 b \right ) {\mathrm e}^{-\frac {\left (-b +\sqrt {b^{2}-4}\right ) t}{2}}+\left (\sqrt {b^{2}-4}\, b^{2}-b^{3}+6 \sqrt {b^{2}-4}+4 b \right ) {\mathrm e}^{\frac {\left (b +\sqrt {b^{2}-4}\right ) t}{2}}+2 \left (b^{3}-4 b \right ) \cos \left (2 t \right )+3 \left (-b^{2}+4\right ) \sin \left (2 t \right )}{4 b^{4}-7 b^{2}-36} \]
✓ Solution by Mathematica
Time used: 0.268 (sec). Leaf size: 88
DSolve[{x''[t]-b*x'[t]+x[t]==Sin[2*t],{x[0]==0,x'[0]==0}},x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {2 e^{\frac {b t}{2}} \left (\frac {\left (b^2+6\right ) \sinh \left (\frac {1}{2} \sqrt {b^2-4} t\right )}{\sqrt {b^2-4}}-b \cosh \left (\frac {1}{2} \sqrt {b^2-4} t\right )\right )+2 b \cos (2 t)-3 \sin (2 t)}{4 b^2+9} \\ \end{align*}