Ordinary Differential Equations/Without x or y
Equations without y
Consider a differential equation of the form
If we can solve for y', then we can simply integrate the equation to get the a solution in the form y=f(x). However, sometimes it may be easier to solve for x. In that case, we get
Then differentiating by y,
Which makes it become
The two equations
is a parametric solution in terms of y'. To obtain an explicit solution, we eliminate y' between the two equations.
If it is possible to express
parametrically as x=f(t), y'=g(t),
then one can differentiate the first equation:
to obtain a parametric solution in terms of t. If it is possible to eliminate t, then one can obtain an integral solution.
Equations without x
Similarly, if the equation
can be solved for y, write y=f(y'). Then the following solution, which can be obtained by the same process as above is the parametric solution:
In addition, if one can express y and y' parametrically
then the parametric solution is
so that if the parameter t can be eliminated, then one can obtain an integral solution.