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JEE Main2013MathematicsDifferential EquationsActual

Consider the differential equation : d y d x = y^3 2 (x y^2-x^2 ) Statement-1: The substitution z=y^2 transforms the above equation into a first order homogenous differential equation. Statement-2: The solution of this differential equation is y^2 e^ -y^2 / x=C .

Options

  1. ABoth statements are false.
  2. BStatement-1 is true and statement- 2 is false.
  3. CStatement-1 is false and statement-2 is true.
  4. DBoth statements are true.

Correct answer

D. Both statements are true.

Step-by-step solution

Given differential equation is d y d x = y^3 2 (x y^2-x^2 ) By substituting z=y^2 , we get diff. eqn. as d z d x = 2 z^2 2 (x z-x^2 ) = z^2 x z-x^2 Now, d x d z = x z - x^2 z^2 = x z [1- x z ] F ( x z ) Hence, statement-1 is true. Now, y^2 e^ -y^2 / x = C satisfies the given diff. equation It is the solution of given diff. equation. Thus, statement -2 is also true.

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