Consider the initial value problem y y2 t y0 y0 1 a Use d

Consider the initial value problem
y\' = y^2 - t; y(0) = y0:
1. (a) Use dfield (Java version) or d eld8 (Matlab version) to plot the direction field of the differential equation with the Display window set for 0 <= t<= 10 and 0 <= y<= 2. Set the window properties to use arrows when displaying the direction field. Print out your plot, it is the answer to this part of the problem.

(b) Use dfield (Java version) or dfield8 (Matlab version) to plot solutions of the initial value problem for 0:8 y0 1:2 with increment of 0.1 (that is y0 =0:8; 0:9; 1:0; 1:1; 1:2).
2. Use forward Euler\'s method with step size h = 0:025, h = 0:05, h = 0:1 and h = 0:2 to find approximate values of the solution on the interval 0 <=t<=5 for y0 = 0:5 and
y0 = 1.

Solution

To solve an intial value probelm we can use the following:

> >y = dsolve(eqn1,\'y(1)=1\',\'t\') y = 1/exp(1/2)*exp(1/2*t^2)