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 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 parametrically as , then one can differentiate the first equation: So that to obtain a parametric solution in terms of . If it is possible to eliminate , 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 can be eliminated, then one can obtain an integral solution.