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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Differential Equations

NDA 2024 · Mathematics · Differential Equations

Paragraph: Let y d x+(x-y^3) d y=0 be a differential equation....

Paragraph: Let ydx+(x−y3)dy=0y d x+\left(x-y^3\right) d y=0 be a differential equation.
Question: What is the solution of the differential equation?

Options

  1. Ay4+2x=cy^4+2 x=c
  2. By4+3x=cy^4+3 x=c
  3. C2xy4+x=c2 x y^4+x=c
  4. D4xy−y4=c4 x y-y^4=c

Correct answer

D. 4xy−y4=c4 x y-y^4=c

Step-by-step solution

Given differential equation is ydxdy+x=y3⇒dxdy+x1y=y2y \frac{d x}{d y}+x=y^3 \Rightarrow \frac{d x}{d y}+x \frac{1}{y}=y^2 (Linear differential equation form dxdy+px=θ\frac{d x}{d y}+p x=\theta )If=\mathrme∫1ydy=yx(I.F)=∫(If)y2dy⇒xy=∫y3dy⇒4xy−y4+cIf =\mathrme^{\int \frac{1}{\mathrm{y}} \mathrm{dy}}=\mathrm{y}x(I . F)=\int(I f) y^2 d y \Rightarrow x y=\int y^3 d y \Rightarrow 4 x y-y^4+c
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