Exercises

(1)
Given that \(y(t)\) satisfies \(y^{\,\prime \prime }+y=2\sin (t)\), \(y(0)=0\), \(y'(0)=0\), determine \(y(2026)\). Round to the nearest tenth.
(2)
Determine the “best guess" for a particular solution to the second order linear nonhomogeneous constant-coefficient differential equation
\(y^{\,\prime \prime }+10y^{\,\prime }+25y=(t+1)e^{-5t}\). The phrase “best guess" here means that there is a particular solution of the form given but no particular solution involving fewer terms.
(a)
\(y_p=Ae^{-5t}\)
(b)
\(y_p=(At+B)e^{-5t}\)
(c)
\(y_p=(At^2+Bt+C)e^{-5t}\)
(d)
\(y_p=Ate^{-5t}\)
(e)
\(y_p=(At^2+Bt)e^{-5t}\)
(f)
\(y_p=(At^3+Bt^2)e^{-5t}\)
(3)
Suppose \(y(t)\) satisfies \(y^{\,\prime \prime }+2y^{\,\prime }+y=2e^{-t}\), \(y(0)=0\), \(y^{\,\prime }(0)=0\). Determine \(y(2)\). Round to the nearest hundredth.
(4)
Suppose \(y(t)\) is the solution to \(y\,^{\prime \prime \prime \prime }-16y=48e^{2t}\) involving the least number of terms. Determine \(y(1)\). Round to the nearest hundredth.

Let \(n,r\) be real numbers, \(r\neq 0\) and \(f(t)\) be a function. Consider the second order linear constant coefficient nonhomogeneous differential equation
\(y\,^{\prime \prime }-2r y\,^{\prime }+r^2 y=f(t) e^{rt}\) on an open interval \(I\) on which the function \(f\) is continuous. Observe that if \(y(t)=u(t)e^{rt}\) for some unknown function \(u(t)\) that is twice differentiable on \(I\) then

\begin{eqnarray*} y\,^{\prime \prime }-2r y\,^{\prime }+r^2 y &=& (u(t)e^{rt})\,^{\prime \prime }-2r (u(t)e^{rt})\,^{\prime }+r^2 u(t)e^{rt} \\ &=& (u\,^{\prime }(t)e^{rt}+ru(t)e^{rt})\,^{\prime }-2r (u\,^{\prime }(t)e^{rt}+ru(t)e^{rt})+r^2 u(t)e^{rt} \\ &=& (u\,^{\prime \prime }(t)e^{rt}+2ru\,^{\prime }(t)e^{rt}+r^2u(t)e^{rt})-2r u\,^{\prime }(t)e^{rt}-2r^2u(t)e^{rt}+r^2 u(t)e^{rt} \\ &=& u\,^{\prime \prime }(t)e^{rt}. \end{eqnarray*}
(5)
Use the above to find the particular solution \(y_p\) to \(y\,^{\prime \prime }-4 y\,^{\prime }+4 y= \sqrt {t}\,e^{2t}\) on \((0,\infty )\) that has the least number of terms. Determine \(y_p(1)\). Round to the nearest hundredth.