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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Differential Equations

NDA 2016 · Mathematics · Differential Equations

What is the curve which passes through the point (1,1) and whose...

What is the curve which passes through the point (1,1)(1,1) and whose slope is 2yx?\frac{2 \mathrm{y}}{\mathrm{x}} ?

Options

  1. ACircle
  2. BParabola
  3. CEllipse
  4. DHyperbola

Correct answer

B. Parabola

Step-by-step solution

dydx=2yx⇒dyy=2dxx\begin{aligned} & \frac{d y}{d x}=\frac{2 y}{x} \\ & \Rightarrow \frac{d y}{y}=2 \frac{d x}{x} \end{aligned} On integration ∫dyy=2∫dxx⇒log⁡y=2log⁡x+log⁡a⇒log⁡y=log⁡x2+log⁡a⇒log⁡y=log⁡(x2⋅a)⇒y=x2a at (1,1);a=1⇒x2=y=4(14)y\begin{aligned} & \int \frac{d y}{y}=2 \int \frac{d x}{x} \\ & \Rightarrow \log y=2 \log x+\log a \\ & \Rightarrow \log y=\log x^2+\log a \\ & \Rightarrow \log y=\log \left(x^2 \cdot a\right) \\ & \Rightarrow y=x^2 a \\ & \text{ at }(1,1) ; a=1 \\ & \Rightarrow x^2=y=4\left(\frac{1}{4}\right) y \end{aligned}\Rightarrow$ the curve is parabola.
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