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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Differential Equations

NDA 2024 · Mathematics · Differential Equations

If (d y)/(d x)=2 e^x y^3, y(0)=(1)/(2) then what is 4 y^2 (2-e^x)...

If dydx=2exy3,y(0)=12\frac{d y}{d x}=2 e^x y^3, y(0)=\frac{1}{2} then what is 4y2(2−ex)equalto?4 y^2 \left(2-e^x\right) equal to?

Options

  1. A1
  2. B2
  3. C3
  4. D4

Correct answer

A. 1

Step-by-step solution

dydx=2exy3⇒dydx3=2exdx\frac{d y}{d x}=2 e^x y^3\Rightarrow \quad \frac{d y}{d x^3}=2 e^x d x Integrating both sides w.r. to x−12y2=2ex+cx-\frac{1}{2 y^2}=2 e^x+c \qquad ....(i) y(0)=12−2=2+c⇒c=−4\begin{aligned} y(0) & =\frac{1}{2} \\ -2 & =2+c \Rightarrow c=-4\end{aligned} The solution is −12y2=2ex−4-\frac{1}{2 y^2}=2 e^x-4⇒4y2(ex)−8y2=−1⇒4y2(2−ex)=1\begin{array}{ll} \Rightarrow & 4 y^2 \left(e^x\right)-8 y^2=-1 \\ \Rightarrow & 4 y^2 \left(2-e^x\right)= 1 \\ \end{array}$
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