$\"\"$

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The equation is $\"\"$ .

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(a) Find that the function is increasing when the condition is $\"\"$ .

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Consider $\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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If $\"\"$ , then $\"\"$ and $\"\"$ .

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Hence, $\"\"$ .

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The value of $\"\"$ when $\"\"$ .

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Therefore, the function is increasing when $\"\"$ .

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Show that the function is decreasing when the condition is $\"\"$ .

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If $\"\"$ , then $\"\"$ and $\"\"$ .

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Hence the product is $\"\"$ .

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Therefore, the function is decreasing when $\"\"$ .

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$\"\"$

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(b)

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The function is $\"\"$ .

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

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Graph the differential equation $\"\"$ . \ \

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Graph :

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$\"\"$

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Observe the graph:

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When, $\"\"$ the vaue of $\"\"$ .

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$\"\"$ the value of $\"\"$ .

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$\"\"$ the value of $\"\"$ .

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Find the equilibrium solutions of the funtion.

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The equilibrium solutions occur when $\"\"$ .

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Substitute $\"\"$ and equate the function to zero.

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$\"\"$

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$\"\"$

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$\"\"$ .

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Equilibrium occurs when $\"\"$ .

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$\"\"$

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(c)

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The function is $\"\"$ \ \

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$\"\"$

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$\"\"$

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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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At the initial condiition $\"\"$ and $\"\"$ .

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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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$\"\"$

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

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$\"\"$

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(d)

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If $\"\"$ ,show that atleast one value of $\"\"$ , the value of $\"\"$ .

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The function is $\"\"$ .

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Equate the function $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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When $\"\"$ , the value of $\"\"$ has atleast a single value for $\"\"$ .

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$\"\"$

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(a) The function is increasing when $\"\"$ and the function is decreasing when $\"\"$ .

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(b)

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$\"\"$

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When $\"\"$ the vaue of $\"\"$ .

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$\"\"$ the value of $\"\"$ .

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$\"\"$ the value of $\"\"$ .

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The Equilibrium conditions are $\"\"$ .

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(c) The solution is $\"\"$ .

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(d) The value of $\"\"$ have atleast a single value for $\"\"$ .