MHTCET 25 Apr 2025 Shift 1 Paper

Concept:The given differential equation can be solved by separating the variables and then integrating both sides.Explanation:Given: $x \frac{dy}{dx} = 2y$.Separating variables: $\frac{dy}{y} = \frac{2dx}{x}$.Integrating both sides: $\int \frac{dy}{y} = \int \frac{2dx}{x}$.$\Rightarrow \log y = 2\log x + \log c$.$\Rightarrow \log y = \log x^2 + \log c$.$\Rightarrow \log y = \log(cx^2)$.$\Rightarrow y = cx^2$.This represents a family of parabolas with vertex at the origin and axis along the Y-axis.Answer:Option B: a family of parabolas with vertex at the origin and axis along the positive Y-axis.