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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Differential Equations

NDA 2018 · Mathematics · Differential Equations

What is the solution of the differential equation ((d y)/(d x))=a...

What is the solution of the differential equation ln⁡(dydx)=ax+by\ln \left(\frac{d y}{d x}\right)=a x+b y ?

Options

  1. Aaeax+beby=ca e^{a x}+b e^{b y}=c
  2. B1aeax+1beby=c\frac{1}{a} e^{a x}+\frac{1}{b} e^{b y}=c
  3. Caeax+be−by=ca e^{a x}+b e^{-b y}=c
  4. D1a\mathrmeax+1b\mathrme−by=c\frac{1}{\mathrm{a}} \mathrme^{\mathrm{ax}}+\frac{1}{\mathrm{ b}} \mathrme^{-\mathrm{by}}=\mathrm{c}

Correct answer

D. 1a\mathrmeax+1b\mathrme−by=c\frac{1}{\mathrm{a}} \mathrme^{\mathrm{ax}}+\frac{1}{\mathrm{ b}} \mathrme^{-\mathrm{by}}=\mathrm{c}

Step-by-step solution

dydx=eax+by=eax⋅eby⇒eaxdx−e−bydy=0Integrating both sides, 1aeax+1be−by=c\begin{aligned} & \frac{d y}{d x}=e^{a x+b y}=e^{a x} \cdot e^{b y} \\ \Rightarrow & e^{a x} d x-e^{-b y} d y=0 \\ & \text{Integrating both sides, } \\ & \frac{1}{a} e^{a x}+\frac{1}{b} e^{-b y}=c\end{aligned}
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