Consider the initial value problem

$y'(t)=f(y(t)),y(0)=a\in \mathbb{R}$

 Which of the following statements are necessarily true?

(a) There exists a continuous function$f:\mathbb{R}\rightarrow \mathbb{R}$  and $a\in\mathbb{R}$ such that the above problem does not have a solution in any neighbourhood of 0.

(b) The problem has a unique solution for every $a\in\mathbb{R}$, when f is Lipschitz continuous.

(c) When is twice continuously differentiable, the maximal interval of existence for the above initial value problem is $\mathbb{R}$.

(d) The maximal interval of existence for the above problem is$\mathbb{R}$, when is bounded and continuously differentiable.

Suppose x : [0,∞) → [0,∞) is continuous and x(0) = 0. If 

$x(t)\le2+\int_0^tx(s)ds,for \;all\:t\ge 0$, for all t ≥ 0,

then which of the following is true?

(a) x( √ 2) ∈ [0, 2]

(b) x( √ 2) ∈ [0, $2e^\sqrt2$]

(c) x( √ 2) ∈ [$\frac{5}{\sqrt2},\frac{7}{\sqrt2}$]

(d) x( √ 2) ∈ [10,∞)

Consider the ODE

$y'(t)=-y^3+y^2+2y,subject\:to\:y(0)=y_0\in(0,2).$

Then $Lim_{t\rightarrow\infty}y(t)$ belongs to

(a) {−1, 0}    (b) {−1, 2}       (c) {0, 2}        (d) {0, +∞}

Consider the ordinary differential equation

y'= y(y − 1)(y − 2). Which of the following statements is true?

(a) If y(0) = 0.5, then y is decreasing.

(b) If y(0) = 1.2, then y is increasing.

(c) If y(0) = 2.5, then y is unbounded.

(d) If y(0) < 0, then y is bounded below.

Assume that $a:[0,\infty)\rightarrow\mathbb{R}$ is a continuous function. Consider the ordinary differential equation:

$y'(x)=a(x)y(x),x>0,y(0)=y_0\neq0.$

Which of the following statements are true?

(a) If $\int_0^\infty|a(x)|dx<+\infty,$ then y is bounded.

(b) If $\int_0^\infty$ |a(x)|dx < +∞, then $Lim_{x\rightarrow\infty}$ y(x) exists.

(c) If $Lim_{x\rightarrow\infty}$ a(x) = 1, then $Lim_{x\rightarrow\infty}$|y(x)| = ∞.

(d) If $Lim_{x\rightarrow\infty}$ a(x) = 1, then y is monotone.

Three solutions of a certain second-order non-homogeneous linear differential equations are

$y_1(x)=1+xe^{x^2},y_2(x)=(1+x)e^{x^2}-1,y_3(x)=1+e^{x^2}$

Which of the following is (are) general solution(s) of the differential equation?

(a) ($c_1$ + 1)y)$_1$+ ($(c_1-c_2)y_2$ − $c_3y_3$, where C$_1$ and C$_2$ are arbitrary constants.

(b) $C_1(y_1-y_2)+C_2(y_2-y_3)$, where C$_1$ and C$_2$ are arbitrary constants.

(c) $C_1(y_1-y_2)+C_2(y_2-y_3)+C3(y_3-y_1)$, where $C_1,C_2\;and\;C_3$  are arbitrary constants.

(d) $C_1(y_1-y_2)+C_2(y_3-y_2)+y_1$, where $C_1\;and\;C_2$are arbitrary constants

 Let u(t) be a continuously differentiable function taking non-negative values for t > 0 satisfying $u'(t)=3u(t)^{\frac{2}{3}}$ and u(0) = 0. Which of the following are possible solutions of the above equation?

(a) u(t) = 0

(b) u(t) = $t^3$

(c) $u(t)= \begin{cases} 0 \; for \; 0<t<1 \\ (t-1)^3 for \;t \ge 3 \end{cases}$

(d) $u(t)=\begin{cases} 0 \; for \; 0<t<1\\ (t-3)^3\; for \;t\ge3 \end{cases}$

Let $y:\mathbb{R}\rightarrow \mathbb{R}$satisfy the initial value problem

$y'(t)=1-y^2(t),t\in\mathbb{R},y(0)=0$

Then

(a) $y(t_1)=1$ for some $t_1\in\mathbb{R}$.

(b) y(t) > −1 for all t ∈ $\mathbb{R}$.

(c) y is strictly increasing in $\mathbb{R}$.

(d) y is increasing in (0, 1) and decreasing in (1,∞).

Consider the initial value problem (IVP) $\frac{dy}{dx}=y^2,y(0)=1,(x,y)\in \mathbb{R} \times \mathbb{R}$ .

Then there exists a unique solution of the IVP on 

(a) (−∞, ∞)

(b) (−∞, 1)

(c) (−2, 2)

(d) (−1,∞)

Consider the initial value problem (IVP) $\frac{dy}{dx}=xy^{\frac{1}{2}},y(0)=0,(x,y)\in \mathbb{R} \times \mathbb{R}$ . Then, which of the following are correct?

(a) The function f(x, y) = $xy^{\frac{1}{3}}$ does not satisfy a Lipschitz condition with respect to y in any neighborhood of y = 0.

(b) There exists a unique solution for the IVP.

(c) There exists no solution for the IVP.

(d) There exists more than one solution of the IVP.

Let$f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with period p > 0. Then $g(x)=\int_x^{x+p}f(t)dt$ is a

(a) constant function

(b) continuous function

(c) continuous function but not differentiable

(d) neither continuous nor differentiable