$\"\"$ \ \

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The differential equation is $\"\"$ .

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Find the solution of the differential equation.

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$\"\"$

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$\"\"$

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Apply integral on each side.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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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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When $\"\"$ close to zero:

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$\"\"$

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$\"\"$ .

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When $\"\"$ is close to zero, $\"\"$ .

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Since $\"\"$ , the graph of the function is constant.

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When $\"\"$ becomes large:

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$\"\"$

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$\"\"$ .

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When $\"\"$ becomes as large, slope of the function becomes very large in magnitude.

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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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$\"\"$

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$\"\"$ .

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

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$\"\"$

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(c)

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Graph the function $\"\"$ , where $\"\"$ is a constant.

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Consider different values of $\"\"$ from $\"\"$ to $\"\"$ .

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$\"\"$

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Observe the graph:

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When $\"\"$ is close to zero, the slope of the function is close to zero.

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When $\"\"$ becomes as large, slope of the function becomes very large in magnitude.

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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 differential equation is $\"\"$ .

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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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$\"\"$ .

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

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$\"\"$ .

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$\"\"$

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(a) When $\"\"$ is close to zero, the slope of the function is close to zero.

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When $\"\"$ becomes as large, slope of the function becomes very large in magnitude. \ \

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

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(c) Graph of the function $\"\"$ , where $\"\"$ is a constant.

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Consider different values of $\"\"$ from $\"\"$ to $\"\"$ .

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$\"\"$

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\ \ When $\"\"$ is close to zero, the slope of the function is close to zero.

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When $\"\"$ becomes as large, slope of the function becomes very large in magnitude. \ \

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(d) $\"\"$ . \ \