General Solution
Initial Condition
Particular Solution


  Differential Equations & Initial Value Problems
A differential equation is an equation involving the derivatives of a function.
To solve a differential equation means you want to find the function itself.
For example, to find the fucntion given that

you integrate with respect to both sides of the equation.

You are taking indefinite integrals, so there will be constants of integration
on the left side and on the right side of the equation.

or

or
.
  is just another constant, simplify the equation by letting

.
Then
.
This the general solution of the differential equation.
The constant can take on any value whatsoever. If we were to graph this
general solution we would have to draw an infinite family of curves, one curve
for each particular value of  .
In other words, the general solution of the differential is not unique, it is not a
single curve but an infinite family of curves.
But if we included an extra condition to the problem of solving the differential
equation - for example, if we required that
         [ This notation means that when , ]
then we are asking for the specific curve, the one curve, of the infinite family of
curves that passes throught the point
.
This extra condition - an initial condition - enables us to determine the one value of  
that solves the problem.
This kind of problem, involving a differential equation and an initial condition is called
an initial value problem.

Evaluate :

so

gives
.
The specific function, or curve, that solves the initial value problem is
.
This is the particular solution of the initial value problem.


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