Ordinary Differential Equations/Without x or y< Ordinary Differential Equations
Equations without yEdit
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 xEdit
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.