Ordinary Differential Equations/Without x or y

< Ordinary Differential Equations

Equations without yEdit

Consider a differential equation of the form

F(x, y')=0.

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

 

and

 

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

F(x, y')=0

parametrically as x=f(t), y'=g(t),

then one can differentiate the first equation:

 

So that

 

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

F(y, y')=0.

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:

y=f(y')

 

In addition, if one can express y and y' parametrically

y=f(t), y'=g(t),

then the parametric solution is

y=f(t),

 

so that if the parameter t can be eliminated, then one can obtain an integral solution.