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

Correct option is C

​​$$\frac{dy}{dt} = y^\frac{5k}{5k+2}, \quad \text{for } t > 0 \\\int y^\frac{5k}{5k+2} \, dy = \int dt \\\ \frac{y^{1-\frac{5k}{5k+2}}}{1-\frac{5k}{5k+2}}= t + C, \\ y^{1-\frac{5k}{5k+2}} = (1-\frac{5k}{5k+2})(t + C) \\y^{\frac{2}{5k+2}} = \frac{2}{5k+2}(t+C) \\y=\big[\frac{2}{5k+2}(t+C)\big]^{\frac{5k+2}{2}}\\\text{Using the initial condition: }\\ y(0) = 0 \ \implies C=0\\\text{Hence, the solution becomes:} \\y=\big[\frac{2}{5k+2}t\big]^{\frac{5k+2}{2}}, \quad \text{for } t > 0 \\\text{Checking continuity and uniqueness:} \\\text{For } t = 0, \; y = 0. \\\text{For } t > 0 , y=\big[\frac{2}{5k+2}t\big]^{\frac{5k+2}{2}}. \\\text{Now, Cauchy problem has 3 choices:} \\1. \text{No solution (not possible)} \\2. \text{Unique solution (not possible, as we have another solution)} \\3. \text{Infinitely many solutions (true)} \\\text{Since } y \text{ is continuous, the correct statement is:} \\\textbf{It has infinitely many solutions which is continuously differentiable on}(0,\infty)$$​​