Exercises

(1)
Evaluate \(\delta (t-1)\ast t^2\) at \(t=2\).
(2)
Suppose \(y(t)\) satisfies the integral equation \(\displaystyle y(t)+\int _0^t y(x)(x-t)\,\mathrm {d}x=1\). Determine \(y(1)\). Round to the nearest thousandth.
(3)
Suppose \(y(t)\) satisfies the integral equation
\(\displaystyle y(t)+\int _0^t (t-x)\,y(x)\,dx=\delta {(t-\pi )}\). Find \(y(4)\) to the nearest tenth.
(4)
Suppose \(y(t)\) satisfies the integral equation \(\displaystyle y(t)+4\int _0^t x^2y(t-x)\,\mathrm {d}x=1\). Determine \(y(1)\). Round to the nearest thousandth.

Hint: \(s^3+a^3=(s+a)(s^2-as+a^2)\).

(5)
Use Euler’s method to estimate \(y(0.6)\), where \(y\) is the unique solution to the IVP \(y\,^{\prime }=1- x^y\), \(y(0)=1\), using a step size of \(\Delta x=0.15\). Round your estimate to the nearest thousandth.