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NDA 2010
Exam
NDA
Year
2010
Subject
Mathematics
Chapter
Differential Equations

NDA 2010 · Mathematics · Differential Equations

What is the solution of the differential equation a(x (d y)/(d x)+2...

What is the solution of the differential equation a(xdydx+2y)=xydydxa\left(x \frac{d y}{d x}+2 y\right)=x y \frac{d y}{d x} ?

Options

  1. Ax2=kyeyax^2=k y e^{\frac{y}{a}}
  2. Byx2=kyeyay x^2=k y e^{\frac{y}{a}}
  3. Cy2x2=kyey2ay^2 x^2=k y e^{\frac{y^2}{a}}
  4. DNone of these

Correct answer

D. None of these

Step-by-step solution

axdydx+2ay=xydydxor dydx=−2ayax−xya−yydy=−2adxxor log⁡y−y=−2alog⁡x+cor log⁡y−y=log⁡x−2a+log⁡kor log⁡y−y=log⁡kx−2a∴log⁡ykx−2a=y or ey=ykx−2a or key=yx2a\begin{aligned} & a x \frac{d y}{d x}+2 a y=x y \frac{d y}{d x} \\ & \text{or } \frac{d y}{d x}=\frac{-2 a y}{a x-x y} \\ & \frac{a-y}{y} d y=-2 a \frac{d x}{x} \\ & \text{or } \log y-y=-2 a \log x+c \\ & \text{or } \log y-y=\log x^{-2 a}+\log k \\ & \text{or } \log y-y=\log k x^{-2 a} \\ & \therefore \log \frac{y}{k x^{-2 a}}=y \text{ or } e^y=\frac{y}{k x^{-2 a}} \text{ or } k e^y=y x^{2 a}\end{aligned}
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