Problem 1 - Applied Calculus

Question

If the solution to \(y^{\prime}=5 x-4 y\) passes through the point \((4,3),\) what is the slope of the solution at that point?

Step-by-Step Solution

Answer

The slope of the solution at the point (4, 3) is 8.

Step 1: Identify the Given Differential Equation

The differential equation given is \(y' = 5x - 4y\). This equation describes the slope of the solution function \(y(x)\) for any point \((x, y)\).

Step 2: Substitute the Given Point into the Equation

We are provided with the point \((4, 3)\). Substitute \(x = 4\) and \(y = 3\) into the differential equation to find the slope of the solution at this specific point.

Step 3: Perform the Calculation

Substitute into the equation: \(y' = 5(4) - 4(3)\). Calculate the slope by evaluating the expression: \(y' = 20 - 12 = 8\).

Step 4: Interpret the Result

The expression evaluates to \(y' = 8\). This means that the slope of the solution at the point \((4, 3)\) is \(8\).

Key Concepts

Slope of Solution

When dealing with a differential equation like \(y' = 5x - 4y\), it's crucial to understand the role of the derivative \(y'\). The derivative represents the "slope" of the solution curve \(y(x)\). At any given point \((x, y)\) on this curve, the slope tells us how steep the line is.
- To find this slope at a particular point, simply substitute the specific \(x\) and \(y\) values into the equation.
Once you've inserted these values, the result is the slope of the solution. In our exercise, at the point \((4, 3)\), after substitution, the slope is calculated to be \(8\).
Understanding the slope helps predict how the solution changes in the small neighborhood around the point. If the slope is high, the function rises or falls steeply. Conversely, a small slope indicates a gentle incline or decline.

Substitution Method

The substitution method is a straightforward technique to find a specific solution to a differential equation when a particular point is known. It involves inserting the values of \(x\) and \(y\) into the differential equation to compute the slope or rate of change at that point.
For our example, given the differential equation \(y' = 5x - 4y\) and the point \((4, 3)\):

• Substitute \(x = 4\) and \(y = 3\) into the equation.
• Replace the variables with numbers: \(y' = 5(4) - 4(3)\).
• Calculate to find the outcome: \(y' = 20 - 12 = 8\).

It's a handy method because it simplifies the task of finding the slope at a specific point without solving the entire differential equation. This approach provides a quick way to understand behavior near specific points.

Point of Intersection

In the context of differential equations, a "point of intersection" typically refers to the point where the solution curve intersects the given coordinate, here the example point was \((4, 3)\). However, in some contexts of algebra and calculus, it may refer to where two functions graphically intersect. In our case, it focuses on where the specific solution of the differential equation passes through the known point.
Understanding this point helps us:- Verify if the calculated slope correctly corresponds to the point.- Utilize this information in constructing or predicting the curve of the solution in practical applications.
In the exercise's case, it is confirmed as \((4, 3)\), meaning that when \(x = 4\), \(y\) is indeed \(3\) on the curve defined by the differential equation, ensuring that our calculations are accurate.