Let \(y:[o,\infty) \rightarrow [0,\infty) \)be a continuously differentiable function satisfying 

\(y(t)=y(0) +\int_0^t y(s)ds,\) for \(t\ge0.\)

Then

(a)   \(y^2(t)=y^2(0)=\int_0^t y^2(s)ds.\)

(b)   \(y^2(t)=y^2(0)+2\int_0^t y^2(s)ds.\)

(c)   \(y^2(t)=y^2(0)+\int_0^t y(s)ds.\)

(d)   \(y^2(t)=y^2(0)+\left(\int_0^ty(s)ds\right)^2 +2y(0)\int_0^ty(s)ds.\)

Let u(t) be a continuously differentiable function taking non-negative values for t > 0 and satisfying \(u'(t)=4u^\frac{3}{4}(t);u(0)=0.\)Then

(a)  u(t)=0.

(b)  \(u(t)=t^4\)

(c)   \(u(t)=\begin{cases} 0\;\;\; for\, 0<t<1,\\ (t-1)^4 \;\;for \;t\ge1 \end{cases} \)

(d)   \(u(t)=\begin{cases} 0\;\;\; for\, 0<t<10,\\ (t-10)^4 \;\;for \;t\ge10 \end{cases} \)

Consider the integral equation

\(y(x)=x^3=\int_0^xsin(x-t)y(t)dt,x\in[0,\pi].\)

Then the value of y(1) is

(a) \(\frac{19}{20}\)       (b) 1       (c) \(\frac{17}{20}\)         (d) \(\frac{21}{20}\)

Let (x(t), y(t)) satisfy the system of ODEs

\(\frac{dx}{dt}=-x+ty,\\ \frac{dy}{dt}=tx-y.\)

If \((x_1(t),y_1(t)) \;and\; (x_2(t),y_2(t))\) are two solutions and\(\Phi(t)=x_1(t)y_2(t)-x_2(t)y_1(t),\), then \(\frac{d\Phi}{dt}\)is equal to

a) \(-2\Phi\)      (b) \(2\Phi\)       (c) \(-\Phi\)         (d) \(\Phi\)

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