Internal problem ID [11462]
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: 1(k).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }+x^{\prime }+x=t^{3}+1-4 \cos \left (t \right ) t} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 52
dsolve(diff(x(t),t$2)+diff(x(t),t)+x(t)=t^3+1-4*t*cos(t),x(t), singsol=all)
\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\left (-4 t +8\right ) \sin \left (t \right )+t^{3}-3 t^{2}-4 \cos \left (t \right )+7 \]
✓ Solution by Mathematica
Time used: 4.528 (sec). Leaf size: 70
DSolve[x''[t]+x'[t]+x[t]==t^3+1-4*t*Cos[t],x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to t^3-3 t^2-4 t \sin (t)+8 \sin (t)-4 \cos (t)+c_2 e^{-t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 e^{-t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )+7 \]