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