Problem 24 - 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.) $$ \frac{d y}{d x}=\frac{1}{2} x^{2} y $$

Step-by-Step Solution

Answer

The function is increasing in the first and second quadrants.

Step 1: Consider the sign of \(x^2\)

Since \(x^2\) is always nonnegative, the sign of the derivative \(\frac{d y}{d x} = \frac{1}{2} x^2 y\) is determined by the sign of \(y\). If \(y\) is positive, the derivative is positive (indicating an increasing function). If \(y\) is negative, the derivative is negative (indicating a decreasing function).

Step 2: Determine the quadrants based on the sign of \(y\)

In the Cartesian coordinate system, \(y\) is positive in the first and second quadrants, hence the derivative would be positive there and the function is increasing in these quadrants. By the same logic, \(y\) is negative in the third and fourth quadrants, hence the derivative would be negative and the function would be decreasing in these quadrants.