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

 The solution of the differential equation\(\frac{dx}{dt}=x^2\) with the initial condition x(0) = 1 will blow up as t tends to

(a) 1       (b) 2        (c) 1/2        (d) ∞

Let \(y_1(x)and \;y_2(x)\) be the solutions of the differential equation \(\frac{dy}{dx}=y+17\) with initial conditions \(y_1(0)=0,y_2(0)=1\). Then

(a) \(y_1\) and \(y_2\) will never intersect.

(b) \(y_1\) and \(y_2\) will intersect at x = 17.

(c) \(y_1\) and \(y_2\) will intersect at x = e.

(d) \(y_1\) and \(y_2\)will intersect at x = 1.

The differential equation

\( \frac{dy}{dx}=60(y^2)^{\frac{1}{5}};x>0,y(0)=0\)

has

(a) a unique solution.

(b) two solutions.

(c) no solution.

(d) infinite number of solutions.

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 \(f\).