Question

Use a MATLAB built-in solver to numerically solve: dy/dx = -yx^2 + 1.5y for 0 lessthanorequalto x lessthanorequalto 3 with y(0) = 2 Make a plot of the numerical solution as a solid red line and the exact solution as green circles as shown. The exact solution is y = 2e^-(2x63 - 9x)/6 Use a MATLAB built-in function to numerically calculate the area of the solution. Use the text command to plot the area with 2-digits of precision as shown.

Answer 1

Solution: For dy/dx yx 2+ 1.5y: func-6(x,y) -y * x^2 + 1.5 * y; using the ode45 built in solver, initial condition y (0)-2 with interval [0, 3] [aValue, yValue ] = ode 45 (func, [0,3], 2); fprintf ('The area of calculation is:\n') %for area with 2-digits precision fprintf ('8.2, %.2f\n",aValue, yValue) %plot the graph with solid red line plot (xValue,yValue,'r-')

Sample Output: The area of calculation is: θ.ee, Θ.13 6.26, θ.39 0.54, Θ.70 0.88, 1.05 1.21, 1.36 1.54, 1.66 1.79, 1.90 2.01, 2.09 2.18, 2.25 2.31, 2.36 2.41, 2.46 2.51, 2.55 2.60, 2.65 2.69, 2.74 2.79, 2.83 2.88, 2.93 2.98, 3.00 2.00, 2.43 2.94, 3.54 4.25, 5.10 5.95, 6.57 6.80, 6.64 5.95, 5.23 4.35, 3.52 2.70, 2.17 1.69, 1.33 1.07, 6.86 6.70, θ.56 8.45, 0.36 6.28 , θ.22 6.17, 6.13 6.18, θ.67 6.05, θ.04 0.63, θ.02

6 0.5 1.5 2.5 For exact solution y = 2en_(2x13-9s)/6: 8x value that is 0< x<-3 x-0:0.1:3; 8y value that is exact solution y 2*exp (-2x.2-9*x) /6; fprintf ('The area of calculation is:n') %for area with 2-digits precision fprintf ( ' % . 2 fin ' , y) plot the graph with qreen circle plot (x,Y, "go-')

Sample Output: The area of calculation is: 6.33 0.13 0.05 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Copyable Code:

For dy/dx = -yx^2 + 1.5y:

func = @(x,y) -y * x^2 + 1.5 * y;
%using the ode45 built in solver, initial condition y(0)=2 with interval [0, 3]
[xValue,yValue] = ode45(func,[0,3],2);
fprintf('The area of calculation is:\n')
%for area with 2-digits precision
fprintf('%.2f, %.2f\n',xValue, yValue)
%plot the graph with solid red line
plot(xValue,yValue,'r-')

For exact solution y = 2e^-(2 x ^3-9x)/6:

%x value that is 0<=x<=3
x=0:0.1:3;
%y value that is exact solution
y=2*exp(-2*x.^3-9*x)/6;
fprintf('The area of calculation is:\n')
%for area with 2-digits precision
fprintf('%.2f\n',y)
%plot the graph with green circle
plot(x,y,'go-')