Problem 23 - Essential Calculus

Question

Determine the quadrants in which the solution of the differential equation is an increasing function. Explain. (Do not solve the differential equation.) $$ \frac{d y}{d x}=\frac{1}{2} x y $$

Step-by-Step Solution

Answer

The solutions of the differential equation are increasing in the 1st and 3rd quadrants.

Step 1: Identify The Variables

The differential equation given is \(\frac{dy}{dx}=\frac{1}{2} xy\). Here, \(\frac{dy}{dx}\) represents the rate of change of the function \(y\) with respect to \(x\), \(x\) is the independent variable and \(y\) is the dependent variable.

Step 2: Determine the Sign of the Derivative

To find where the function \(y\) is increasing, we need to find the areas where the derivative (represented by the right-hand side of the equation, \(\frac{1}{2}xy\)) is positive. The product of \(x\) and \(y\) will be positive in the 1st quadrant (where both \(x\) and \(y\) are positive) and in the 3rd quadrant (where both \(x\) and \(y\) are negative).

Step 3: Final Conclusion

From the previous step, we determined that the solution is increasing in the 1st and 3rd quadrants. Thus, these are the quadrants in which the solution to the given differential equation is an increasing function.