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The general solution of the differential equation (3 y-7 x+7) d x+(7 y-3 x+3) d y=0 is

Options

  1. A(x-y+1)^2(x+y-1)^5=C
  2. B(x+y+1)^5(x-y-1)^2=C
  3. C(x-y-1)^2(x+y-1)^5=C
  4. D(x+y-1)^7=C

Correct answer

C. (x-y-1)^2(x+y-1)^5=C

Step-by-step solution

We have, (3 y-7 x+7) d x+(7 y-3 x+3) d y=0 (7 x-3 y-7) d x+(3 x-7 y-3) d y=0 ( i ) Given equation is non-homogeneous and a₁ b₂-a₂ b₁=-40 0 On solving 7 x-3 y-7=0 and 3 x-7 y-3=0 we get x=1, y=0 Now, substitude x=1+u and y=0+v=v d x=d u and d y=d v in Eq. (i), we get (7 u-3 v) d u+(3 u-7 v) d v=0 ( ii ) This homogeneous equation in u and v . Put, u=t v d u=t d v+v d t in Eq. (ii), we get (7 t v-3 v)(t d v+v d t)+(3 t v-7 v) d v=0(7 t-3)(t d v+v d t)+(3 t-7) d v=0 (7 t^2-7 ) d v+v(7 t-3) d t=0 d v v + 7 t-3 7 t^2-7 d

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