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AP EAMCET20228 Jul 2022Evening ShiftMathematicsDifferential EquationsActual

Suppose that f(x, y) and g(x, y) are homogeneous functions of same order. If x=V y reduces the equation d y d x = f(x, y) g(x, y) to the form d V d y = 1 y (F(V)) , then F(V)=

Options

  1. A( f(1, V) g(1, V) -V )
  2. B( f(V, 1) g(V, 1) -V )
  3. C( g(1, V) f(1, V) -V )
  4. D( g(V, 1) f(V, 1) -V )

Correct answer

D. ( g(V, 1) f(V, 1) -V )

Step-by-step solution

d y d x = f(x, y) g(x, y) ...(i) x=V y array ll & d x d y =V+y d v d y & y d V d y = d x d y -V & d V d y = 1 y ( d x d y -V ) array From Eq. (i), d x d y = g(x, y) f(x, y) = g(V y, y) f(V y, y) = g(V, 1) f(V, 1) aligned & d V d y = 1 y ( g(V, 1) f(V, 1) -V ) & F(V)= ( g(V, 1) f(V, 1) -V ) aligned

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