answer:To derive the first integral, multiply the second-order equation by the first derivative of the dependent variable, integrate, and isolate the first derivative term to obtain an expression in terms of the dependent variable and a constant. This expression represents the level surfaces of the first integral, which can be used to create a phase portrait using contour plots.

answer:Using the identity {eq}cos(arcsin(x))= sqrt {1-x^2}}{/eq}, we have: sqrt {1-(5x)^2} Simplifying the expression under the square root: sqrt {1-25x^2} Therefore, the simplified form of the expression is {eq}sqrt {1-25x^2}}{/eq}.

answer:To find the least common multiple (LCM) of -16 and 8, we first note that the LCM of two numbers remains the same even if one or both numbers are negative. Thus, we can find the LCM of the absolute values, which are 16 and 8. The prime factorization of 16 is (2^4) (since (16 = 2 times 2 times 2 times 2)), and for 8, it is (2^3) (since (8 = 2 times 2 times 2)). To find the LCM, we take the highest power of each prime factor. In this case, the highest power of 2 is (2^4). Therefore, the LCM of 16 and 8 is (2^4 = 16). So, the LCM of -16 and 8 is 16.

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