Question

Q7. Solve the initial value problem $x'(t) = Ax(t) + \begin{pmatrix} e^{-t} \\ -e^t \end{pmatrix}$, $x(0) = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$ where $x \in C^1([0, T]; \mathbb{R}^2)$ and $A = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}$. (3 marks) Q8. Prove the second shift theorem for Laplace transforms. That is show that for a function $f : [0, \infty) \to \mathbb{R}$ of exponential order and piecewise continuous we have $\mathcal{L}[t \to H(t - c)f(t - c)](s) = e^{-cs}\mathcal{L}[f](s)$. for all $c > 0$ constant such that the Laplace transforms in the above relation make sense. (3 marks) Q9. Consider the BVP given by $y''(t) + 36ay(t) = 0$, $y(0) = 1$ and $y(\pi/6) = 1$. Find all constants $a \in \mathbb{R}$ such that the BVP has a unique solution. (2 marks)

          Q7. Solve the initial value problem
$x'(t) = Ax(t) + \begin{pmatrix} e^{-t} \\ -e^t \end{pmatrix}$, $x(0) = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$
where $x \in C^1([0, T]; \mathbb{R}^2)$ and
$A = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}$.
(3 marks)
Q8. Prove the second shift theorem for Laplace transforms. That is show that for a function $f : [0, \infty) \to \mathbb{R}$ of
exponential order and piecewise continuous we have
$\mathcal{L}[t \to H(t - c)f(t - c)](s) = e^{-cs}\mathcal{L}[f](s)$.
for all $c > 0$ constant such that the Laplace transforms in the above relation make sense.
(3 marks)
Q9. Consider the BVP given by $y''(t) + 36ay(t) = 0$, $y(0) = 1$ and $y(\pi/6) = 1$.
Find all constants $a \in \mathbb{R}$ such that the BVP has a unique solution.
(2 marks)

Q7. Solve the initial value problem
x'(t) = Ax(t) +
< p m a t r i x >, x(0) =
< p m a t r i x >
where x ∈ C^1([0, T]; ℝ^2) and
A =
< p m a t r i x >.
(3 marks)
Q8. Prove the second shift theorem for Laplace transforms. That is show that for a function f : [0, ∞) →ℝ of
exponential order and piecewise continuous we have
ℒ[t → H(t - c)f(t - c)](s) = e^-csℒ[f](s).
for all c > 0 constant such that the Laplace transforms in the above relation make sense.
(3 marks)
Q9. Consider the BVP given by y”(t) + 36ay(t) = 0, y(0) = 1 and y(π/6) = 1.
Find all constants a ∈ℝ such that the BVP has a unique solution.
(2 marks)

Added by Linda E.

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