Problem 28 - Essential Calculus

Question

Solve the differential equation. Use a graphing utility to graph three solutions, one of which passes through the given point. $$ \frac{d y}{d x}=\frac{2 x}{x^{2}-9}, \quad(0,4) $$

Step-by-Step Solution

Answer

The solution to the differential equation is \(y = \ln |x^{2} - 9| + 4 - \ln|9|\).

Step 1: Separation of Variables

The given differential equation is \(\frac{d y}{d x}=\frac{2 x}{x^{2}-9}\). Here, the function can be separated into variables on different sides of the equation. This can be done in the form \(\frac{dy}{dx} = g(x)h(y)\) where \(g(x) = \frac{2x}{x^{2} - 9}\) and \(h(y) = 1\). It's recognizable as a separable equation of the form \(\frac{dy}{dx} = g(x) h(y)\). Therefore, we rewrite it as \(dy = \frac{2x}{x^{2} - 9} dx\).

Step 2: Integration

Both sides of the equation can now be integrated. \(\int dy = \int \frac{2x}{x^{2} - 9} dx\). By integrating, one gets \(y = \int \frac{2x}{x^{2} - 9} dx = \ln |x^{2} - 9| + C\), where C is the constant of integration.

Step 3: Find the Constant C

To find the constant of integration, C, utilise the initial condition specified by the problem, which is that the solution should pass through the point (0,4). So, substitute \(x = 0\) and \(y = 4\) into the solution: \(4 = \ln |0^{2} - 9| + C\). From this, \(C = 4 - \ln|9|\) is obtained.

Step 4: Final Solution

With \(C = 4 - \ln|9|\), the full solution to the differential equation can be written as \(y = \ln |x^{2} - 9| + 4 - \ln|9|\). The graph can now be plotted with 3 random values of \(x\). Remember to calculate \(y\) for each chosen \(x\) and plot these points on the graph.