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NDA 2012
Exam
NDA
Year
2012
Subject
Mathematics
Chapter
Differential Equations

NDA 2012 · Mathematics · Differential Equations

What is the general solution of the differential equation x^2 d...

What is the general solution of the differential equation x2dy+y2dx=0x^2 d y+y^2 d x=0 ?

Options

  1. Ax+y=cx+y=c
  2. Bxy=cx y=c
  3. Cc(x+y)=xyc(x+y)=x y
  4. DNone of the above where cc is the constant of integration

Correct answer

C. c(x+y)=xyc(x+y)=x y

Step-by-step solution

x2dy+y2dx=0x2dy=−y2dx\begin{aligned} & x^2 d y+y^2 d x=0 \\ & x^2 d y=-y^2 d x\end{aligned}\frac{d x}{x^2}+\frac{d y}{y^2}=0onintegrating,wegeton integrating, we get-\frac{1}{x}-\frac{1}{y}=\mathrm{C}_1wherewhere\mathrm{C}_1isconstantofintegrationis constant of integrationx+yxy=−C1x+y=−C1(xy)\begin{aligned} \frac{x+y}{x y} & =-\mathrm{C}_1 \\ x+y & =-\mathrm{C}_1(x y)\end{aligned}⇒1−C1(x+y)=xy⇒C(x+y)=xy\Rightarrow \frac{1}{-\mathrm{C}_1}(x+y)=x y\Rightarrow \quad \mathrm{C}(x+y)=x y \quad where C=1−C1\mathrm{C}=\frac{1}{-\mathrm{C}_1}
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