Determining the Solution to the Differential Equation: xy²y’
The differential equation given is in the form xy²y’ = 0. To solve this, let’s break it down step by step:
Step 1: Understand the Equation
This equation implies that the product of x, y², and the derivative y’ is equal to zero. For the product to hold true, at least one of the factors must be zero:
- x = 0
- y² = 0
- y’ = 0
Step 2: Analyze the Factors
Let’s consider each factor:
- x = 0: This indicates that the solution includes the vertical line at x = 0.
- y² = 0: This implies that y = 0. Thus, we have a solution along the horizontal axis, which states that for any value of x, y can equal zero.
- y’ = 0: This means that the derivative of y is zero, indicating that y is a constant function. Therefore, if y = C (a constant), this would satisfy the equation as well.
Step 3: Conclusion
Based on the analysis above, we can conclude that the functions y = 0 and y = C are solutions to the differential equation xy²y’ = 0. The straightforward nature of this equation leads us to these simple constant solutions, which are valuable in understanding different behaviors of the system described by the differential equation.
Therefore, any constant function or the function where y equals zero would meet the criteria laid out by the differential equation.