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AP EAMCET2016MathematicsDifferential Equations

The solution of the differential equation (2 x-4 y+3) d y d x +(x-2 y+1)=0 is ( C is an arbitrary constant)

Options

  1. A[(2 x-4 y)+3]=x-2 y+C
  2. B[2(2 x-4 y)+3]=2(x-2 y)+C
  3. C[2(x-2 y)+5]=2(x+y)+C
  4. D[4(x-2 y)+5]=4(x+2 y)+C

Correct answer

D. [4(x-2 y)+5]=4(x+2 y)+C

Step-by-step solution

Given differential equation, aligned & (2 x-4 y+3) d y d x +(x-2 y+1)=0 & d y d x = (x-2 y+1) 2(x-2 y+3) (i) aligned Let, x-2 y=v 1-2 d y d x = d y d x 1 2 (1- d y d x )= d y d x Now, substitute v=x-2 y and d y d x in eq. (i) integration both the sides, we get aligned & 1 2 (1- d v d x )=- ( v+1 2 v+3 ) & d v d x =1+ 2 v+2 2 v+3 & 2 v+2 4 v+5 d v=d x & 1 2 4 v+6 4 v+5 d v= d x & 1 2 ( 4 v+6 4 v+5 + 1 4 v+5 ) d v= d x & 1 2 v+ 1 2 4 [4 v+5]=x+C & 4 v+ [4 v+5]=8 x+C & 4(x-2 y)+ [4(x-2 y)+5]=8 x+C & [4(x-2 y)+5]=4(x+2

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