​$$\text{The extremal of the functional} \\J(y) = \int_0^1 \left[ 2(y')^2 + xy \right] dx, \quad y(0) = 0, \; y(1) = 1, \; y \in C^2[0,1] \\\text{is}$$​​

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Correct option is D

​$$J(y) = \int_0^1 \left[ 2(y')^2 + xy \right] dx, \quad y(0) = 0, \; y(1) = 1 \\\text{Using Euler's equation: }\\ \frac{\partial f}{\partial y} - \frac{d}{dx} \left( \frac{\partial f}{\partial y'} \right) = 0 \\x - \frac{d}{dx} \left( 4y' \right) = 0\\ \implies x - 4y'' = 0\\\text{(using integration)}\\ \implies y'' = \frac{x}{4} \\y' = \frac{x^2}{8} + a, \quad y = \frac{x^3}{24} + ax + b \\\text{Now, using boundary conditions: } y(0) = 0 \implies b = 0, \quad y(1) = 1 \implies 1 = \frac{1^3}{24} + a(1) + 0\\ \implies a = 1 - \frac{1}{24} = \frac{23}{24} \\\textbf{Hence, the solution is: } \\y = \frac{x^3}{24} + \frac{23}{24}x$$​​