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NDA 2 2025 Maths, question 93

Topic: Differential equations

Q93

Consider the following statements :

I. y=xe2xy=xe^{2x} is the solution of dydx=y(2+1x)\frac{dy}{dx}=y\left( 2+\frac{1}{x} \right).

II. y=xln⁡∣x∣+cxy=x\ln |x|+cx is the solution of dydx=x+yx\frac{dy}{dx}=\frac{x+y}{x}.

Which of the statements given above is/are correct?

  1. (a) I only
  2. (b) II only
  3. (c) Both I and IICorrect answer
  4. (d) Neither I nor II

Solution.

Answer: (c) Both I and II

  1. For I, differentiating xe²ˣ gives e²ˣ(1+2x)=y(2+1/x).
  2. For II, differentiating x ln|x|+cx gives ln|x|+1+c=(x+y)/x.
  3. Both satisfy their equations on intervals not containing x=0.