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NDA 2010
Exam
NDA
Year
2010
Subject
Mathematics
Chapter
Differential Equations

NDA 2010 · Mathematics · Differential Equations

What is the general solution of (1+x^x) y d y=e^x d x ?

What is the general solution of \left(1+x^x\right) y d y=e^x d x ?

Options

  1. Ay2=ln⁡[c2(ex+1)]2y^2=\ln \left[c^2\left(e^x+1\right)\right]^2
  2. By=ln⁡[c(ex+1)]y=\ln \left[c \left(e^x+1\right)\right]
  3. Cy2=ln⁡[c(ex+1)]y^2=\ln \left[c \left(e^x+1\right)\right]
  4. DNone of the above where cc is a constant of integration.

Correct answer

A. y2=ln⁡[c2(ex+1)]2y^2=\ln \left[c^2\left(e^x+1\right)\right]^2

Step-by-step solution

ydy=∫exex+1dx∴y22=log⁡(ex+1)+log⁡cy2=2log⁡(ex+1)+2log⁡cy2=log⁡(ex+1)2c2\begin{aligned} & y d y=\int \frac{e^x}{e^{x+1}} d x \\ & \therefore \frac{y^2}{2}=\log \left(e^x+1\right)+\log c \\ & y^2=2 \log \left(e^x+1\right)+2 \log c \\ & y^2=\log \left(e^x+1\right)^2 c^2\end{aligned}
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