Problem 5
Solve the integral equation $$ f(t)-\frac{1}{6} \int_{0}^{t}(t-y)^{3} f(y) d y=t^{2} . $$
Problem 6
Evaluate the integrals (a) \(\int_{0}^{2}\left(4-x^{2}\right)^{\frac{3}{2}} d x\), (b) \(\int_{0}^{\infty} \frac{1-\cos x}{x^{2}} d x\).
Problem 6
Let \(f\) and \(g\) be continuously differentiable and such that \(f, g, f^{\prime}\), and \(g^{\prime}\) are absolutely integrable on the interval \([0, \infty)\), and \(f(0)=g(0)=0\). Prove that $$ (f * g)^{\prime}=f^{\prime} * g=f * g^{\prime} . $$
Problem 7
Compute the inverse Laplace transform of the functions (a) \(\int_{0}^{t}(t-y)^{2} \cos 2 y d y\), (b) \(\int_{0}^{t} \sin (t-y) \cos y d y\).
Problem 8
Solve the differential equation $$ y^{\prime \prime}+2 y^{\prime}+2 y=\sin a t, \quad y(0)=0, \quad y^{\prime}(0)=0, $$ where \(a\) is a given constant.
Problem 9
Find the inverse Laplace transform of \(\frac{1}{(s+1)^{2}\left(s^{2}+4\right)}\).
Problem 10
Solve the following systems of equations:
(a)
$$
\left\\{\begin{array}{l}
x^{\prime}(t)+2 x(t)-4 y(t)=f(t) \\
y^{\prime}(t)+x(t)-2 y(t)=0 \\
x(0)=y(0)=0
\end{array}\right.
$$
$$
\text { where } f(t)=\left\\{\begin{array}{ll}
t, & 0 \leq t \leq 1, \\
1, & 1
Problem 11
Use the Laplace transform to solve the following systems of partial
differential equations:
(a)
$$
\begin{cases}u_{x}+2 x u_{t}=2 x, & 0
