how do i solve differential equation dy/dx=3/(2-x) and use graphing utility to graph 3 solutions,one which passes through the point(1,0)

Integrate

dy = 3 dx/(2-x)

y - yo = -3 ln (2-x)+- 3 ln(2-xo)

y = -3 ln[(2-x)/1] = -3 ln (2-x)

To solve the differential equation dy/dx = 3/(2-x), we can use separation of variables. The step-by-step process is as follows:

Step 1: Separate the variables.
- Multiply both sides of the equation by dx to isolate dy.
dy = (3/(2-x)) dx

Step 2: Integrate both sides.
- Integrate both sides with respect to their respective variables.
∫dy = ∫(3/(2-x)) dx

- For the left side, the integral of dy is simply y.
y = ∫(3/(2-x)) dx

Step 3: Evaluate the integral.
- To integrate (3/(2-x)), we can use a substitution.
- Let u = 2-x, then du = -dx.
- Substitute these values into the integral.
y = ∫(-3/u) du

- The integral becomes ∫(-3/u) du = -3 ln|u|, where ln represents the natural logarithm.
y = -3 ln|u| + C (C is the constant of integration)

Step 4: Re-substitute u back as 2-x.
y = -3 ln|2-x| + C (C is the constant of integration)

Step 5: Find the particular solution passing through the point (1,0).
- Plug in x = 1 and y = 0 into the equation.
0 = - 3 ln|2-1| + C
0 = -3 ln|1| + C
0 = -3(0) + C
C = 0

- Therefore, the particular solution passing through (1,0) is given by:
y = -3 ln|2-x|

To graph three solutions, including the one passing through (1,0), you can use any graphing utility like Desmos or GeoGebra. Here's how you can do it using Desmos:

1. Open a web browser and go to www.desmos.com.
2. Click on the "Start Graphing" button to open the graphing calculator.
3. In the input box, enter "y = -3 ln|2-x|" to plot the solution that passes through (1,0).
4. To plot additional solutions, you can modify the constant of integration (C) in the equation. For example, you can try C = 1 and C = -1.
- Enter "y = -3 ln|2-x| + 1" to plot a solution for C = 1.
- Enter "y = -3 ln|2-x| - 1" to plot a solution for C = -1.
5. Adjust the viewing window to see the graph clearly by zooming in or out.
6. The graphing utility will display the graph of the differential equation with three solutions passing through (1,0) for different values of C.

To solve the given differential equation dy/dx = 3 / (2 - x), you can follow these steps:

Step 1: Separate the variables.
Multiply both sides of the equation by (2 - x) to isolate dy: (2 - x) dy = 3 dx.

Step 2: Integrate both sides.
Integrate each side of the equation with respect to their respective variables:
∫ (2 - x) dy = ∫ 3 dx.

The left-hand side becomes: y - (1/2)x^2 + C1, where C1 is the constant of integration.

The right-hand side becomes: 3x + C2, where C2 is another constant of integration.

Therefore, the equation becomes: y - (1/2)x^2 + C1 = 3x + C2.

Step 3: Rewrite the equation in slope-intercept form.
Rearrange the equation to get it in the form y = mx + b:
y = (1/2)x^2 + 3x + (C1 - C2).

Step 4: Find the particular solution that passes through (1, 0).
Substitute the coordinates of the point (1, 0) into the equation y = (1/2)x^2 + 3x + (C1 - C2):
0 = (1/2)(1)^2 + 3(1) + (C1 - C2).
Simplify the equation: 0 = 1/2 + 3 + (C1 - C2).
Combine like terms: 0 = 7/2 + (C1 - C2).

Since we need a particular solution, we can assume that C1 - C2 = -7/2.

Substitute this value back into the equation to get the particular solution:
y = (1/2)x^2 + 3x - 7/2.

Now, to graph three solutions in a graphing utility that passes through the point (1, 0), follow these steps:

Step 1: Open a graphing utility program.

Step 2: Enter the equation y = (1/2)x^2 + 3x - 7/2 into the graphing utility.

Step 3: Set the x-axis and y-axis ranges to show the region around the point (1, 0).

Step 4: Graph the equation.

By following these steps, you will obtain a graph that includes the particular solution passing through the point (1, 0) and two other solutions to the given differential equation.