Question

Use Euler's method with step size0.4 to estimatey(2) , where y(x) is the solution of the initial-value problem y'=4x+y^2, y(0)=1.

. y(2)=

Answer 1

Just plug in the values given. Below is a table showing iteration number, (x,y) and next y value. Looks like y(2) = 405.0307

0: (0.0000,1.0000) -> 1.4000
1: (0.4000,1.4000) -> 2.8240
2: (0.8000,2.8240) -> 7.2940
3: (1.2000,7.2940) -> 30.4949
4: (1.6000,30.4949) -> 405.0307
5: (2.0000,405.0307) -> 66028.1757

Looks like y grows pretty fast. As is to be expected, since y' grows like y^2.

Answer 2

Use Euler's method with step size0.4 to estimate y(2) , where y(x) is the solution of the initial-value problem y'=5x+y^2, y(0)=1.

Y(1)=