Consider the differential equation Y' = -3x^2y, y(0)=3A) using Euler's method ( by hand ),...

Question

Consider the differential equation Y' = -3x^2y,
y(0)=3
A) using Euler's method ( by hand ), approximate the solution y(1/2) using a step size h= 0.1
B) Using the Runge-Kutta method ( by hand), approximate the solution y (1/2) using a step size h=0.25
C) find the exact solution to the differential equation and use it to find y(1/2)
NB: I need please stey by step solution.
Also please make sure you do this question to the best of your knowledge. Last time I handed in four questions, 15 days ago. All the answers were incorrect that is why I am sending you this notice inorder not to happen again.
Thank you.

David · Accepted Answer

Sol: (a) 
 
     
     
2
0 0
1 0 0 0
2
2 1 1 1
3 2
We are given 0,  3,   , 3 . From Euler's method,
                 
                 ,  3 0.1 3 0 3 3
                 ,  3 0.1 3 0.1 3 2.991
                 
dy
x y f x y x y
dx
y y hf x y
y y hf x y
y y
    
       
       
       
     
     
2
2 2
2
4 3 3 3
2
5 4 4 4
,  2.991 0.1 3 0.2 2.991 2.9551
                 ,  2.9551 0.1 3 0.3 2.9551 2.8753
                 ,  2.8753 0.1 3 0.4 2.8753 2.7373
Euler's method thus yields the appro
hf x y
y y hf x y
y y hf x y
     
       
       
 ximation  0.5 2.7373.y 

       
       
       
2
1
21
2 1
22
3 2
4
Sol: (b) step 1:
               , 0.25 3
               , 0.25 3 0.125 0.

Consider the differential equation Y' = -3x^2y, y(0)=3 A) using Euler's method ( by hand ), approximate the solution y(1/2) using a step size h= 0.1 B) Using the Runge-Kutta method ( by hand),...

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