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

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

Let (X, d) be your favourite metric (X, d). How does open ball in (X, d) look like ?

Visualize the open ball B(f, R) in \(\left(C[0,1],d_{\infty}\right)\), where f is the identity function. We say that Y ⊆ X is open in X if for every y ∈ Y, there exists r > 0 such that B(y, r) ⊆ Y, that is,

{z ∈ X : d(z, y) < r} ⊆ Y.

Give five of your favourite open subsets of \(\mathbb{R^2}\) endowed with any of your favourite metrics.

Give five of your favourite non-open subsets of \(\mathbb{R^2}\) .