This is the method most commonly known simply as "Euler's Method" (yes you could just Google it but I'm being nicer than that): http://mathworld.wolfram.com/EulerForwardMethod.html
If the differential equation is of the form y'=f(x,y), h is the step-size, (x0,y0) is the initial condition, and (x_n,y_n) is the nth approximation, then x_(n+1)=x_n+h, and y_(n+1)=y_n+h*f(x_n,y_n).
Below I will show you how to calculate the approximations using Euler's Method, using h=dt=5 (this problem uses t instead of x, and dt instead of h); when getting the later solutions, you will get the earlier solutions.
When t=5, n=t/h=1, and when t=20, n=t/h=4; this means we will need y1 and y4 for the first two solutions (you can figure out the rest).
f(t,y)=-1/14*(y-70), t0=0, and y0=201.
Then y1=y0+h*f(t0,y0)=201+5*(-1/14*(201-70))=2159/14, and t1=t0+h=0+5=5.
You can keep going like this, but it's probably easier to use a spreadsheet.
You can enter this into Excel or LibreOffice with a formula like this (it's easy in this case because f(t,y) does not actually depend on t), to more easily get the solutions; after entering 201 into A1, 0 into B1, and 5 into C1, place the formula =B1+C$1 into B2, place =A1+C$1*(-1/14)*(A1-70) into A2, and then select A2 and B2 and drag down to get the other solutions (look for the rows where column B has values 5, 20, 30, and 60). When you've read them off, you can change C1 to 1 and it will re-calculate the approximations using Euler's Method.
10/18/2014 5:58:37 AM