Consider the initial value problem  (IVP)

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​$$\text{Let } y_0 > 0, \ z_0 > 0, \ \alpha > 1. \\[10pt]\text{Consider the following two differential equations:} \\[10pt](*) \quad \frac{dy}{dt} = y^{\alpha} \quad \text{for} \ t > 0, \quad y(0) = y_0 \\[10pt](**) \quad \frac{dz}{dt} = -z^{\alpha} \quad \text{for} \ t > 0, \quad z(0) = z_0 \\[10pt]\text{We say that the solution to a differential equation exists globally if it exists for all } t > 0.$$

Which of the following statements is true?

For a positive integer n, let $$f^{(n)}$$​ denote the n-th derivative of f.

Suppose anentire function f satisfies: $$f^{(2)}+f=0$$​​

Which of the following is correct?

​$$\text{Let } f : \mathbb{R}^2 \to \mathbb{R} \text{ be continuous and} \\[10pt]f(t, x) < 0 \quad \text{if } tx > 0, \\[10pt]f(t, x) > 0 \quad \text{if } tx < 0. \\[10pt]\text{Consider the problem of solving the following:} \\[10pt]\dot{x} = f(t, x), \quad x(0) = 0. \\[10pt]\text{Which of the following is true?}$$​​

​$$\text{Consider the ODE: }\\t y' - 3y = t^2 y^{1/2}, \quad y(1) = 1.\\\text{Find the value of } y(2).$$​​

​$$\text{Consider the differential equation:} \\[10pt]x^2 y'' - 2x(x + 1)y' + 2(x + 1)y = 0. \\[10pt]\text{If a polynomial is a solution, then the degree of the polynomial is equal to:}$$​​

​$$\text{For } \lambda \in \mathbb{R}, \text{ consider the system of differential equations:} \\x'_1 = x_1 + 2x_2 + 2x_3, \\x'_2 = 2x_2 + x_3, \\x'_3 = -x_3 + 2x_2 + \lambda x_3. \\[10pt]\text{If } \vec{x}(t) = \vec{a} \, t e^{2t} \, (\text{for some } \vec{a}) \text{ is a solution of the system, then the value of } \lambda \text{ is equal to.}$$​​

​$$\text{Let } k \text{ be a positive integer. Consider the differential equation} \\\begin{cases} \frac{dy}{dt} = \frac{5k}{y^{5k+2}}, & \text{for } t > 0, \\y(0) = 0 \end{cases} \\\text{Which of the following statements is true?}$$​​

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​$$\text{Let } y_0 > 0, \ z_0 > 0, \ \alpha > 1. \\[10pt]\text{Consider the following two differential equations:} \\[10pt](*) \quad \frac{dy}{dt} = y^{\alpha} \quad \text{for} \ t > 0, \quad y(0) = y_0 \\[10pt](**) \quad \frac{dz}{dt} = -z^{\alpha} \quad \text{for} \ t > 0, \quad z(0) = z_0 \\[10pt]\text{We say that the solution to a differential equation exists globally if it exists for all } t > 0.$$

Which of the following statements is true?