Internal problem ID [10437]
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(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }+x^{\prime }+x-t^{2} {\mathrm e}^{3 t}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 47
dsolve(diff(x(t),t$2)+diff(x(t),t)+x(t)=t^2*exp(3*t),x(t), singsol=all)
\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\frac {\left (169 t^{2}-182 t +72\right ) {\mathrm e}^{3 t}}{2197} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 55
DSolve[x''[t]+x'[t]+x[t]==t^2*exp(3*t),x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to 3 \exp \left ((t-3) t^2+6\right )+e^{-t/2} \left (c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \\ \end{align*}