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 $\mathbb{R^2}$ .

Let B[0, 1] denote the set of all bounded functions $f:[0,1]\rightarrow \mathbb{R}$ endowed with the metric $d_{\infty}$. Show that C[0, 1] can not be open in B[0, 1].

Show that the open unit ball in $(C[0,1],d_{\infty})$ can not be open in (C[0, 1], $d_1$), where 

$d_1(f,g)=\int_{[0,1]}|f(t)-g(t)|dt$.