Final answer:
To sketch the slope field, calculate the slopes at multiple points and plot arrows to indicate the direction and magnitude. To sketch the solution passing through a point, follow the direction of the arrows in the slope field and adjust the y-coordinate accordingly.
Step-by-step explanation:
To sketch the slope field for the differential equation, we can use a grid of points and calculate the slope at each point. The slope at a given point (x, y) can be found by evaluating the equation dy/dx = 5y using the coordinates of that point. For example, if we choose (1,2) as a point on the grid, the slope at that point would be dy/dx = 5(2) = 10.
Once we have calculated the slopes for multiple points on the grid, we can plot arrows indicating the direction and magnitude of the slope at each point. This will give us the slope field.
Next, to sketch the solution passing through the point (0,1), we can use the slope field as a guide. We start at the given point and follow the direction of the arrows in the slope field. At each step, we move in the direction of the slope and adjust the y-coordinate accordingly. By repeating this process, we trace the curve that represents the solution to the differential equation passing through the given point.