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