If y = ( x) - x^2 , then d^2y dx^2 + 4x dy dx + 4x^2y =

Step-by-step solution

Given y = ( x) - x^2 Differentiating with respect to x : 1 y dy dx = x - 2x dy dx = y( x - 2x) Differentiating again with respect to x : d^2y dx^2 = dy dx x - y ^2 x - 2x dy dx - 2y The given expression is d^2y dx^2 + 4x dy dx + 4x^2y . Substituting d^2y dx^2 into the expression: d^2y dx^2 + 4x dy dx + 4x^2y = ( dy dx x - y ^2 x - 2x dy dx - 2y ) + 4x dy dx + 4x^2y = dy dx x - y ^2 x + 2x dy dx - 2y + 4x^2y = dy dx ( x + 2x) - y ^2 x - 2y + 4x^2y Substituting dy dx = y( x - 2x) : = y( x - 2x)( x + 2x) - y ^2 x - 2y