1. Verify that the given function is a solution of the given differential equation. Show all relevant steps. Then find the particular solution corresponding to the initial value y(0) =2 . (2% points) 1 3 y=, Yy =Xy \\)C.)C2 2. Verify that the given function is a solution of the given differential equation. Show all relevant steps. (24 points) y:xe3x, y'=6y'+9y=0 3. Match each differential equation to its slope field. Write the number of the corresponding slope field on the blank line. (2 points) a. y '= xy+x+y+1 c. y' = xy - y- +1 b. y' = x + sin (y) d. y' = x(y - 1)2 Slope Field 1 Slope Field 2 2 . -1 - -2 1 -3 -3 Slope Field 3 Slope Field 4 1 - 0 0 - -1 - -2 -2 -3 4. Suppose that y' = y + . If a particular solution passes through (0, -2) , is y increasing, decreasing, or neither at that point? How do you know? (1 point)5. The function with the given graph is a solution to one of the following differential equations. Decide which is the correct equation. Summarize how you made your decision. (11/2 points) A. y' = 1+ xy B. y' = - 2xy C. y' = 1 - 2xy 6. Use Euler's method with step size h = 0.2 to estimate the value of y (2) , where y is the solution of the initial value problem y' = y + 2x , y (1) = 1 . Show the inputs of the calculations. (21/2 points) n Xn Yn Yn+ 1 = Y, + h[f '(xn, yn)] Extra Credit (1 point each) A. Use Sage to generate a table of 4" order Runge-Kutta approximations with step size h = 0.2 to estimate the value of y (2) , where y is the solution of the initial value problem y' = y + 2x , y (1) = 1. Copy and paste your code into an email and send it to me at (julia.partlow@pcc.edu) B. Use Sage to produce the slope field for y ' = > >(y - 4) and the solution curves for the initial conditions y (0) = -1, y(0) =2, y(0) = 5 , all on the same set of axes. Use different colors for each solution curve and use display settings xmin = -5, xmax = 5, ymin = -3, ymax = 7. Copy and paste your code into an email and send it to me (julia.partlow@pcc.edu)