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There is a language with 12 distinct letters. How many words that are less than four letters long can possibly be formed? (Assum
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The general form of a solution of the differential equation is already provided for us:
where . We now want to find a solution
such that
and
. Therefore, all we need to do is find the constants
and
that satisfy the initial conditions. For the first condition, we have:
For the second condition, we need to find the derivative first. In this case, we have:
Therefore:
This means that we must solve the following system of equations:
If we add the equations above, we get:
If we now substitute into either of the equations in the system, we get:
This means that the solution obeying the initial conditions is:
Indeed, we can see that:
which do correspond to the desired initial conditions.
If you don't know the first derivative of , but you do for
and
, you can derive the former via the quotient rule:
or if you know the derivative of :
As for the second derivative, you can use the power/chain rules:
or if you don't know the derivative of ,
which is the same as the previous result since