Exercises

(1)
Determine \(y(2026)\) given that \(y(t)\) satisfies \(y^{\,\prime }+y=t+e^{2026}\delta (t-1)\), \(y(0)=0\). Round to the nearest tenth.
(2)
Suppose \(y(t)\) satisfies the IVP \(4y\,^{\prime \prime }+4y\,^{\prime }+y=5\delta (t-2)\), \(y(0)=0\), \(y\,^{\prime }(0)=1\). Determine \(y(4)\). Round to the nearest hundredth.
(3)
Determine \(y(2026)\) given that \(y(t)\) satisfies \(y^{\,\prime \prime }=\delta (t-\pi )\), \(y(0)=0\), \(y'(0)=1\). Round to the nearest tenth.
(4)
Suppose \(y(t)\) satifies the IVP \(\displaystyle y^{\,\prime \prime }=\sum _{n=0}^{\infty }\delta (t-n)\), \(y(0)=0\), \(y^{\,\prime }(0)=0\). Use Laplace transforms to determine \(y(\pi )\). Round to the nearest thousandth.
(5)
There exist positive constants \(a,k\) so the solution to the IVP
\(y^{\,\prime \prime }+2y^{\,\prime }+2y=k\delta \left (t-a\right )\), \(y(0)=0\), \(y^{\,\prime }(0)=1\) satisfies \(y=0\) for all \(t\geq a\). Determine \(k\) such that \(a\) is minimal. Round to the nearest hundredth.