Solving Initial Value Problems with Mathematica's Solver
Hopefully you recall asking Mathematica to give the syntax of its
DSolve command in an earlier laboratory exercise. We're going
to be using the command again today, so it would be a good idea to be reminded
of the details. I'll save you the trouble of asking Mathematica
yourself—here's the result we got before:
?DSolve
Notice the third form of the command shown: it describes how to solve a
list of equations.
Now consider a very simple initial value problem—one that you could
practically solve in your head:
dy/dx = 2x, y(0) = 5.
Clearly, direct integration gives the general solution:
y = x^{2} + C
and the initial condition soon yields C = 5, so the solution to the
initial value problem is:
y = x^{2} + 5
But, look at the initial value problem again! Isn't it really just a list of
two equations? (A very short list, admittedly.) So the initial value problem,
in a sense, fits the second form of the DSolve command that we
read above. In other words, to solve an initial value problem we
simply use DSolve with a list of equations, the first of which is the
differential equation itself, and the remaining equations in the list being the
initial conditions.
Go ahead and solve the initial value problem above,
using Mathematica, with the command:
DSolve[{y'[x]==2x,y[0]==5},y[x],x]
You
can switch to Mathematica by clicking on the button at left. This will
open up a fresh notebook for you. Remember that it will take it a while to start
up! Don't forget to come back here when you're done! See you in a few
minutes.
Welcome back! Let's move on to discuss your result...

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