UPSC NDA I 2022 Mathematics Solved Paper

Concept:The given equation is a separable first-order differential equation. It can be solved by separating variables, integrating, and then applying the initial condition to find the constant.Explanation:We start with $\frac{dy}{dx} = (\ln 5)y$. Separate the variables: $\frac{dy}{y} = (\ln 5)\,dx$. Integrate both sides: $\int \frac{dy}{y} = \int (\ln 5)\,dx$. This gives: $\ln y = x\ln 5 + C$, where $C$ is the integration constant. Use logarithm properties: $\ln y = \ln 5^x + C$. Rewrite as: $\ln y = \ln (5^x) + \ln C$ (by absorbing $C$ into $\ln C$). Thus: $\ln y = \ln (C \cdot 5^x)$. Exponentiate both sides: $y = C \cdot 5^x$. Now apply the initial condition $y(0) = \ln 5$. Substitute $x=0$: $\ln 5 = C \cdot 5^0 = C \cdot 1$, so $C = \ln 5$. Therefore: $y = (\ln 5) \cdot 5^x$. To find $y(1)$, set $x=1$: $y(1) = (\ln 5) \cdot 5^1 = 5 \ln 5$.Answer:$5 \ln 5$ Thus, the correct option is D.