Suppose that we use Euler's method to approximate the solution to the differential equation
dydx=x4y;y(0.5)=7.dydx=x4y;y(0.5)=7.
Let f(x,y)=x4/y.f(x,y)=x4/y.
We let x0=0.5x0=0.5 and y0=7y0=7 and pick a step size h=0.2.h=0.2. Euler's method is the the following algorithm. From xnxn and yn,yn, our approximations to the solution of the differential equation at the nth stage, we find the next stage by computing
xn+1=xn+h,yn+1=yn+h⋅f(xn,yn).xn+1=xn+h,yn+1=yn+h⋅f(xn,yn).
Complete the following table. Your answers should be accurate to at least seven decimal places. nn xnxn ynyn
00 0.50.5 77
11 .7 7.0018
22 .9 33 1.1 44 1.3 55 1.5 The exact solution can also be found using separation of variables. It is
y(x)=y(x)= 0