Examples:

1. Consider this simple ODE that can be solved symbolically.

             y'=2t, y(0)=1
           

   The solution is

   We can find the symbolic solution in two ways, by hand with integration methods and with Maple.

   By hand solution

   
   integrate both sides
   
   
   use the initial condition to find C
   

   Symbolic ODE Solution in Maple

   1. Enter the ode.
   2. Use dsolve to solve the ODE.

   As shown here, the result is a symbolic expression for the function y(t).

   

2. Or, we can find a numeric solution by using Euler's Method.

   Euler's Method

   Choose h=0.2 and solve from to .

   k	0	1	2	3	4	5	6
   	0	0.2	0.4	0.6	0.8	1.0	1.2
   	1	1	1.08	1.24	1.48	1.80	2.20

   In the figure below the approximate (numeric) solution produced by Euler's method is compared to the exact solution.

   

   Notice that the numeric solution (circles) deviates substantially from the exact solution (blue curve). We can improve the accuracy of the numeric solution by reducing the value of the time step h. By doing so, the finite difference more accurately approximates the actual derivative