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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Differential Equations

NDA 2022 · Mathematics · Differential Equations

If (d y)/(d x)=( n 5) y with y(0)= 5 , then what is y(1) equal to?

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

Options

  1. A0
  2. B5
  3. C2ln⁡52 \ln 5
  4. D5ln⁡55 \ln 5

Correct answer

D. 5ln⁡55 \ln 5

Step-by-step solution

dydx=(ln⁡5)y⇒∫1ydy=(ln⁡5)∫dxln⁡y=(ln⁡5)x+cput x=0,y=ln⁡5ℓln⁡(ln⁡5)=0+c⇒c=ln⁡(ln⁡5)∴ln⁡y=(ln⁡5)x+ln⁡(ln⁡5)put x=1ℓny=ln⁡5+ln⁡(ln⁡5)=ln⁡(5ln⁡5)y=5ln⁡5\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}
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