The first order system of equations equivalent to

\(x''+p(t)x'+q(t)x=0\)

is

\(\frac{dX}{dt} =A(t)X with \; X= {\begin{pmatrix} x_1\\x_2\end{pmatrix} },\)and \(\frac{dX}{dt}=\begin{pmatrix} x'_1\\x'_2 \end{pmatrix},\)

where A(t) is the \(2\times2 matrix\)

(a)  \(\begin{pmatrix} 1&0\\-p&-q \end{pmatrix}\)   (b)\(\begin{pmatrix}0&1\\-q&-p \end{pmatrix}\)(c)\(\begin{pmatrix} 0&1\\q&p \end{pmatrix}\)   (d) \(\begin{pmatrix}-p&-q \\1&0\end{pmatrix}\)

The function f(x, y) = x|y|,(x, y) ∈ R × R with respect to the y variable

(a) satisfies a Lipschitz condition on the strip −1 ≤ x ≤ 1, −∞ < y < ∞.

(b) satisfies a Lipschitz condition on the strip −∞ < x < ∞, −1 ≤ y ≤ 1.

(c) does not satisfy a Lipschitz condition on −1 ≤ x ≤ 1, −1 ≤ y ≤ 1.

(d) does not satisfy a Lipschitz condition at any point on R × R.

According to the excavated evidence, the domestication of animal began in
(A) Lower Palaeolithic period
(B) Middle Palaeolithic period
(C) Upper Palaeolithic period
(D) Mesolithic period

The initial value problem \(\frac{dy}{dx}=x^2+y^2,y(0)=0\), in the neighborhood of (0, 0) has

(a) no solution

(b) a singular solution

(c) a unique differential solution

(d) infinite number of solutions

The largest interval on which the solution of the initial value problem \(\frac{dx}{dt}=x^3\) , x(0) = 1 is defined as

(a) \(t>\frac{1}{2}\)      (b) \(t<\frac{1}{2}\)       (c) \(\frac{-1}{2}<t<\frac{1}{2}\)           (d) −1 < t < 0

The initial value problem \(2x\frac{dy}{dx}=3(2y-1),y(0)=\frac{1}{2}\) has

(a) a unique solution

(b) more than one, but only finitely many solutions

(c) infinitely many solutions

(d) no solution

The differential equation \(y''+2x(y')^2=0\), satisfying the condition y(1) = 0, y'(1) = 1

(a) has no solution

(b) has a unique solution

(c) has two distinct solutions

(d) has an infinite number of solutions

The ordinary differential equation \(x\frac{dy}{dx}-y=2x^2\) with initial conditions y(0) = 0 has (a) no solution

(b) a unique solution

(c) two distinct solutions

(d) an infinity of solutions

Consider the differential equation \(y'=y^2-1\), x > 0 together with initial condition y(0) = \(y_0\) Then for −1 < \(y_0\) < 1, all the solutions y(x) are such that:

(a) y → −∞ as x → ∞.

(b) y → −1 as x → ∞.

(c) graph of y(x) is concave downwards.

(d) graph of y(x) is concave upwards.

Give five of your favourite metrics on \(\mathbb{R^2}\).

Show that C[0, 1] is a metric space with metric \(d_{\infty}(f,g):=||f-g||_{\infty}\). 

An open ball in a metric space (X, d) is given by

 \(B_d(x,R):=\{ {y \in X:d(y,x)<R }\}\)