Q222 / 270 · NDA 2022 (Phase 1) · Mathematics

If $\frac{d y}{d x}=(\ell \operatorname{n} 5) y$ with $y(0)=\ln 5$, then what is $y(1)$ equal to?

  • A
    0
  • B
    5
  • C
    $2 \ln 5$
  • Correct answer
    D
    $5 \ln 5$

Solution

$\begin{aligned} & \frac{d y}{d x}=(\ln 5) y \Rightarrow \int \frac{1}{y} d y=(\ln 5) \int d x \\ & \ln y=(\ln 5) x+c \\ & \text {put } x=0, y=\ln 5 \\ & \ell \ln (\ln 5)=0+c \Rightarrow c=\ln (\ln 5) \\ & \therefore \ln y=(\ln 5) x+\ln (\ln 5) \\ & \text {put } x=1 \\ & \ell n y=\ln 5+\ln (\ln 5)=\ln (5 \ln 5) \\ & y=5 \ln 5\end{aligned}$

Hence, the correct answer is option: 'D'.

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