Consider the initial value problem $y'(t)=f(t)y(t),y(0)=1,$ where $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous. Then this initial value problem has

(a) indefinitely many solutions for some $f$.

(b) a unique solution in $\mathbb{R}$.

(c) no solution in $\mathbb{R}$ for some $f$.

(d) a solution in the interval containing 0, but not on $\mathbb{R}$ for some $f$.

Consider the equation

$\frac{dy}{dx}=(1+f^2(t))y(t),y(0)=1:t\ge 0,$

where f is a bounded continuous function on [0, ∞). Then

(a) The equation admits a unique solution y(t) and further $Lim_{t\rightarrow \infty}y(t)$ exists and is finite

(b) The equations admits two linearly independent solutions

(c) This equation admits a bounded solution for which $Lim_{t\rightarrow \infty}y(t)$ does not exist

(d) The equation admits a unique solution y(t) and further, $Lim_{t\rightarrow \infty}y(t)=\infty$

Consider the differential equation

$\frac{dy}{dx}=y^2,(x,y)\in\mathbb{R}\times\mathbb{R}$

Then

(a) all solutions of the differential equation are defined on $(-\infty,\infty)$.

(b) no solution of the differential equation are defined on $(-\infty,\infty)$.

(c) the solution of the differential equation satisfying the initial condition $y(x_0)=y_0,y_0>0,$is defined on $(-\infty,x_0+\frac{1}{y_0})$

(d) the solution of the differential equation satisfying the initial condition $y(x_0)=y_0,y_0>0$, is defined on  $(x_0-\frac{1}{y_0},\infty)$.

The initial value problem

$\dot{\mathbf{x}}(t)=3x^{\frac{2}{3}},x(0)=0$

in an interval around t = 0, has

(a) no solution

(b) a unique solution

(c) finitely many linearly independent solutions

(d) infinitely many linearly independent solutions

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