Question

Consider the solution of the differential equation y′=3y passing through y(0)=0.5. Sketch the slope field for this differential equation, and sketch the solution passing through the point (0,0.5). Use Euler's method with step size h=0.2 to estimate the solution at x=0.2,0.4,…,1, using these to fill in the following table. Note: Be sure not to round your answers at each step! help (numbers) Plot your estimated solution on your slope field. Compare the solution and the slope field. Is the estimated solution an over or under estimate for the actual solution? A. over B. under Check that y=0.5e3x is a solution to y′=3y with y(0)=0.5.

Answer 1

To sketch the slope field and estimate the solution of a differential equation, we can use Euler's method with a given step size. The actual solution can then be compared with the estimated solution to determine if it is an over or under-estimate. In this case, the actual solution is y = 0.5 * e^(3x) which satisfies the differential equation y' = 3y and initial condition y(0) = 0.5.

Step-by-step explanation:

To sketch the slope field for the given differential equation, we need to find the slope at various points on the x-y plane. For the equation y' = 3y, the slope at any point (x, y) is 3y. We can choose several points on the x-y plane, calculate the slope at each point, and then plot arrows with those slopes.

Using Euler's method with a step size of h=0.2, we can estimate the solution at x=0.2, 0.4, ..., 1. Starting with the initial condition y(0) = 0.5, we can use the formula y(x+h) = y(x) + h * f(x, y(x)) to calculate the estimates.

Finally, we can plot the estimated solution on the slope field and compare it with the actual solution. Based on the given differential equation y' = 3y, the actual solution is given by y = 0.5 * e^(3x). We can check that this equation satisfies the given differential equation and initial condition.