Question

In this Section, you should attempt TWO OUT OF THE THREE QUESTIONS. Write down your solutions in the notebook(s) provided, making sure all your work is clearly visible. 7. (a) Find in implicit form the solution $y = y(x)$ of the following initial value problem: $$(\cos y) \frac{dy}{dx} = 2x + x \sin y, \quad y(0) = 0.$$ [13 Marks] (b) Consider the following system: $$(S): \begin{cases} -x + 2py + I & 7z = -3 \\ 2x - 3py & 9z = 5 \\ -2x + 2py + (p^2 + 2p + 1)z & = p - 5 \end{cases}$$ where $x$, $y$ and $z$ are unknowns, and $p$ is a parameter. Find all the values of $p$, if any, for which $(S)$ has no solution, and infinitely many solutions respectively. Solve $(S)$ in the case $p = -2$. [12 Marks]

          In this Section, you should attempt TWO OUT OF THE THREE QUESTIONS.
Write down your solutions in the notebook(s) provided, making sure all your
work is clearly visible.
7. (a) Find in implicit form the solution $y = y(x)$ of the following initial value problem:
$$(\cos y) \frac{dy}{dx} = 2x + x \sin y, \quad y(0) = 0.$$
[13 Marks]
(b) Consider the following system:
$$(S): \begin{cases} -x + 2py + I & 7z = -3 \\ 2x - 3py & 9z = 5 \\ -2x + 2py + (p^2 + 2p + 1)z & = p - 5 \end{cases}$$
where $x$, $y$ and $z$ are unknowns, and $p$ is a parameter.
Find all the values of $p$, if any, for which $(S)$ has no solution, and infinitely many
solutions respectively.
Solve $(S)$ in the case $p = -2$.
[12 Marks]

In this Section, you should attempt TWO OUT OF THE THREE QUESTIONS.
Write down your solutions in the notebook(s) provided, making sure all your
work is clearly visible.
7. (a) Find in implicit form the solution y = y(x) of the following initial value problem:

(cos y) (dy)/(dx) = 2x + x sin y, y(0) = 0.

[13 Marks]
(b) Consider the following system:

(S): -x + 2py + I 7z = -3
2x - 3py 9z = 5
-2x + 2py + (p^2 + 2p + 1)z = p - 5

where x, y and z are unknowns, and p is a parameter.
Find all the values of p, if any, for which (S) has no solution, and infinitely many
solutions respectively.
Solve (S) in the case p = -2.
[12 Marks]

Added by Michael B.

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