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NDA 2010
Exam
NDA
Year
2010
Subject
Mathematics
Chapter
Differential Equations

NDA 2010 · Mathematics · Differential Equations

The growth of a quantity N(t) at any instant t is given by (d...

The growth of a quantity N(t)N(t) at any instant tt is given by dN(t)dt=αN(t)\frac{d N(t)}{d t}=\alpha N(t). Given that N(t)=cekj,cN(t)=c e^{k j}, c is a constant, what is the value of α\alpha ?

Options

  1. Ac
  2. Bkk
  3. Cc+kc+k
  4. Dc−kc-k

Correct answer

D. c−kc-k

Step-by-step solution

dN(t)dt=αN(t)∴log⁡N(t)=αt+c⇒log⁡N(t)=eαt+c or cekt=eαt+cor kt=αt+c or αt=kt−c,α=k−c/t\begin{aligned} & \frac{d \mathrm{ N}(t)}{d t}=\alpha \mathrm{N}(t) \\ & \therefore \log \mathrm{N}(t)=\alpha t+c \\ & \Rightarrow \log \mathrm{ N}(t)=e^{\alpha t+c} \text{ or } c e^{k t}=e^{\alpha t+c} \\ & \text{or } k t=\alpha t+c \text{ or } \alpha t=k t-c, \alpha=k-c / t\end{aligned}
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