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AP EAMCET2005MathematicsDifferential Equations

If x^2 y-x^3 d y d x =y^4 x , then x^3 y is equal to

Options

  1. Ax
  2. B2 x+c
  3. C-3 x+c
  4. D3 x+c

Correct answer

C. -3 x+c

Step-by-step solution

Given that x^2 y-x^3 d y d x =y^4 x On dividing by y^4 , we get aligned x^2 y^3 - x^3 y^4 d y d x & = x x^3 y^4 d y d x & = x^2 y^3 - x aligned array cc & 1 y^4 d y d x - 1 x y^3 =- 1 x^3 x Let & - 1 y^3 =t array array llrl & 1 y^4 d y d x & = 1 3 d t d x & & 1 3 d t d x + 1 t x & = 1 x^3 x & & d t d x + 3 x t & = 3 x^3 x array This is a linear differential equation in t , on comparing with d t d x +P t=Q , we get gathered P= 3 x , Q= 3 x^3 x I.F. =e^ P d x =e^ 3 x d x =e^ x^3 =x^3 gathered Complete solution is ali

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