$\"\"$

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The differential equation is $\"\"$ .

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(a)

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If the derivative of the function $\"\"$ , then the function $\"\"$ is zero or decreasing on that interval.

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$\"\"$ .

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The solution of the differential equation must be zero or decreasing.

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$\"\"$

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(b)

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The solution of the differential equation is $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$ .

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Therefore, the differential equation solution is $\"\"$ .

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$\"\"$

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(c)

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The family of solutions differential equations are $\"\"$ .

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Consider the solution of the differential equation $\"\"$ is $\"\"$ .

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The solution $\"\"$ is not in the member of $\"\"$ .

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$\"\"$

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(d) Find the solution of the differential equation satisfies the initial condition $\"\"$ .

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The solution of the differential equation is $\"\"$ .

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The initial condition is $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ .

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$\"\"$

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(a) The solution of the differential equation must be zero or decreasing.

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(b) The differential equation solution is $\"\"$ .

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(c) The solution $\"\"$ is not in the member of $\"\"$ .

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(d) $\"\"$ .