we are given that \\( \\frac { d y } { d x } = x ^ { 2 } - 2 y \\).\nfind an expression for \\( \\frac { d ^…

we are given that \\( \\frac { d y } { d x } = x ^ { 2 } - 2 y \\).\nfind an expression for \\( \\frac { d ^ { 2 } y } { d x ^ { 2 } } \\) in terms of \\( x \\) and \\( y \\).\n\\( \\frac { d ^ { 2 } y } { d x ^ { 2 } } = \\)

we are given that \\( \\frac { d y } { d x } = x ^ { 2 } - 2 y \\).\nfind an expression for \\( \\frac { d ^ { 2 } y } { d x ^ { 2 } } \\) in terms of \\( x \\) and \\( y \\).\n\\( \\frac { d ^ { 2 } y } { d x ^ { 2 } } = \\)

Answer

Explanation:

Step1: Differentiate both sides of the equation

Differentiate (\frac{dy}{dx}=x^{2}-2y) with respect to (x). Using the sum - difference rule ((u - v)^\prime=u^\prime - v^\prime) (where (u = x^{2}) and (v = 2y)), and the power rule ((x^{n})^\prime=nx^{n - 1}) ((n = 2) for (x^{2})) and the chain rule ((y)^\prime=\frac{dy}{dx}). (\frac{d^{2}y}{dx^{2}}=\frac{d}{dx}(x^{2})-\frac{d}{dx}(2y)) (\frac{d^{2}y}{dx^{2}} = 2x-2\frac{dy}{dx})

Step2: Substitute (\frac{dy}{dx})

Since (\frac{dy}{dx}=x^{2}-2y), substitute it into the above - equation. (\frac{d^{2}y}{dx^{2}}=2x - 2(x^{2}-2y))

Step3: Simplify the expression

Expand the right - hand side: (\frac{d^{2}y}{dx^{2}}=2x-2x^{2}+4y)

Answer:

(2x - 2x^{2}+4y)