1)Let's check, if "c_1y_1" is a solution:
"c_1y_1=2c_1(x+1)." .
Let's check, if "2c_1(x+1)" is equal to "x*(2c_1(x+1))'+\\cfrac{((2c_1(x+1))')^2}{2}" :
"x*(2c_1(x+1))'+\\cfrac{((2c_1(x+1))')^2}{2} = 2c_1x+2c_1^2" and it's not equal to "2c_1(x+1)" for arbitary "c_1" so it's not a solution.
2)Let's check if "c_2y_2" is a solution:
"c_2y_2=\\cfrac{-c_2x^2}{2}" .
Let's check, if "\\cfrac{-c_2x^2}{2}" is equal to "x*(\\cfrac{-c_2x^2}{2})'+\\cfrac{((\\cfrac{-c_2x^2}{2})')^2}{2}:"
"x*(\\cfrac{-c_2x^2}{2})'+\\cfrac{((\\cfrac{-c_2x^2}{2})')^2}{2}=-c_2x^2+\\cfrac{c_2^2x^2}{2}" and it's not equal to "\\cfrac{-c_2x^2}{2}" for arbitary "c_2" so it's not a solution.
3)Let's check, if "y_1+y_2" is a solution:
"(y_1+y_2)'=(\\cfrac{-x^2}{2}+2x+2)'=2-x" .
So "xy\u00b4+\\cfrac{( y\u00b4)\u00b2}{2}=2x-x^2+\\cfrac{1}{2}" and it's not equal to "y_1+y_2" .
Answer: none of the three expressions is a solution.
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