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To reduce the differential equation d y d x + P x . y = Q x . y n to the linear form, the substitution is:

Options

  1. Av = 1 y n
  2. Bv = 1 y n - 1
  3. Cv = y n
  4. Dv = y n - 1

Correct answer

B. v = 1 y n - 1

Step-by-step solution

The given differential equation is d y d x + P x y = Q x . y n ⇒ 1 y n . d y d x + y - n + 1 P x = Q ( x ) Put 1 y n - 1 = v ⇒   - n + 1 . y - n . d y d x = d v d x ⇒   1 ( - n + 1 ) . d v d x + P x . v = Q ( x ) ⇒ d v d x + 1 - n P x v = 1 - n Q ( x ) Hence, required substitution is v = 1 y n - 1 .

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