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AP EAMCET201920 Apr 2019Morning ShiftMathematicsDifferential EquationsActual

If (c ) is a parameter, then the differential equation of the family of curves (x^2=c(y+c)^2 ) is

Options

  1. A(x ( d y d x )^3+y ( d y d x )^2-1=0 )
  2. B(x ( d y d x )^3-y ( d y d x )^2+1=0 )
  3. C(x ( d y d x )^3+y ( d y d x )^2+1=0 )
  4. D(x ( d y d x )^3-y ( d y d x )^2-1=0 )

Correct answer

D. (x ( d y d x )^3-y ( d y d x )^2-1=0 )

Step-by-step solution

Given equation of the family of curves ( aligned & x^2=c(y+c)^2 & x= c (y+c) (i) aligned ) [on taking square root both sides] Now, differentiate both sides w.r.t. (x ), we get ( aligned I & = c d y d x c & = d x d y (ii) aligned ) From Eqs. (i) and (ii) on eliminating ' (c ) ', we get ( aligned & x= ( d x d y ) [y+ ( d x d y )^2 ] & x ( d y d x )^3=y ( d y d x )^2+1 & x ( d y d x )^3-y ( d y d x )^2-1=0 aligned ) Hence, option (4) is correct.

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