91Ó°ÊÓ

A constant solution in the context of differential equations means that the function remains unchanged over time. This essentially leads to the derivative of the function being zero. For the differential equation given, which is \( y' = y^2 + 4 \), we are asked if there could be a constant solution. To determine this, we set \( y' = 0 \). This implies \( y^2 + 4 = 0 \). However, if you think about it, \( y^2 \) represents the square of a real number, which is always zero or positive—never negative, zero at the least. Thus, adding any non-zero positive value, such as 4, will always give a positive result, never zero. Consequently, \( y^2 + 4 = 0 \) has no real solutions, meaning there are no constant solutions possible for the differential equation \( y' = y^2 + 4 \).

• No constant value of \( y \) can satisfy the equation.
• The sum of a non-negative number and 4 cannot equal zero.

The concept of relative extrema is closely related to the behavior of a function's graph. They occur where the slope of the tangent to the curve—represented by the derivative—is zero. In other words, these points are where the graph reaches a local maximum or minimum. For the equation \( y' = y^2 + 4 \), we previously explained that \( y' \) can never be zero due to the nature of \( y^2 + 4 \) being always positive for real numbers. This absence of zero in the derivative means no flattening of the curve occurs anywhere. Therefore, the function doesn't possess relative maxima or minima.

• Relies on \( y' = 0 \) to identify extrema.
• Never zero, hence no relative extrema in the graph of \( y = \phi(x) \).

Analyzing the behavior of a solution's graph from a differential equation can provide insights into the nature of the function across a range of input values. Differential equations frequently inform us how quickly or slowly a solution changes. In the case of \( y' = y^2 + 4 \), since \( y' \) is always positive, it indicates that as \( x \) increases or decreases, \( y \) will generally move upward, continuously increasing. This is because the derivative does not change sign to indicate any downtrend; it remains positive, indicating a strictly increasing function for all real \( y \). Consider the following:

• Derivative \( y' \) always positive suggests a rising solution.
• No zero derivative, implying no peaks or valleys—steady increase only.
• The lack of constant solutions implies the graph is never steady.

Understanding graph behavior is crucial when discussing the dynamics that differential equations describe.