It is a very unusual thing to be given a differential equation that will just, well, integrate. Usually some manipulations must be performed, whether it is simplifying, grouping like terms, simplifying, making substitutions or separating variables – the technique illustrated here. In general a differential equation may have
and
terms on both sides, but if the equation is of a certain form –
- we can rearrange to have all terms including
on the right hand side and all terms includingon the left hand side, obtaining in this case,
We can then integrate both sides:
Example: Solve the differential equation
Multiply by
and divide by
to give
We can now integrate:
is a product which we integrate by parts obtaining
To find the constant
we need what is called a boundary condition. Suppose then that we have that when
Substitute these values into (1) to obtain
hence
Example: Solve
Factorise the right hand side into
to give
which is separable.
Multiply by
and divide byto give
We can now integrate:
(1)
If we are to makethe subject we exponentiate both sides, raising
to the power of both sides:
where
Notice how the constant term
in (1) becomes the constant factor
when we exponentiate.