Arbitrary constants in the solution of an ODE

Question

I have some fundamental doubts regarding the arbitrary (or undetermined) constants present in the solution of an ODE.

1) Is it always true that the order of the DE is equal to the number of arbitrary constants in its solution? If yes,is there any formal proof of it?(DE by Krantz says that its always true but gives no proof).If true would it also imply that an 'nth' order ODE has 'n' independent solutions?Some insights are in this link : Is it true that the number of arbitrary constants in the solution always equal to order of the ordinary differential equation? but still does not make it very much clear

2)Can the arbitrary constant take any possible values? please refer to this link here : Can an arbitrary constant in the solution of a differential equation really take on any value? Need some clear explanation here.

3)Is it always possible to form a differential equation by eliminating the arbitrary constants if both the above statements are true,i.e,it can take any possible arbitrary value & that the number of arbitrary constants is equal to the order of the ODE?

Here are the links to the questions which substantiate my doubts:

1)How to solve implicit differential equation? The question here adresses that we can form an ODE which obeys the above rules and also re-solve the ODE to reach the given solution.

2)Find the differential equation of the given primitive This link gives a question where we can eliminate the arbitrary constant to get the ODE by solving a quartic.But that would significantly increase the difficulty level to re-solve the ODE to get at the solution.

3)forming ODE by elimination of arbitrary constant This question is becoming extremely hard for me to even get at the ODE let alone resolving it backwards to get the solution.

I seriously need clarification about this basic concepts without proceeding further in DEs.

Answer 1

Now you are studying DE so you must know integration is opposite of differentiation.

1)It is true as the arbitrary constants are introduced through integeration so are removed by diffrentiation. 2 arbitary constant means two integeration means two time diffrentiation means the order is 2.

2)arbitary constant can take on any value for which you can define x and y. why i am saying this is because arbitary constant represents a family of curve. i.e. for every one value of arbitary constant there exists a curve. so lets say with the arbitary constant a the curve is (x-2)^2 + (y-3)^2 = a^2 a circle with centre (2,3) and radius a. now a cant be 0 or say any complex number.

3)yes all you have to do is either find the values of arbitary constants in terms of (derivatives,x,y....)and substitute or alienate the arbitary constant (i.e. no x or y or any derivative is multiplied to it) and take derivative to eliminate the arbitary constant. (x-2)^2 + (y-3)^2 = a^2 now a^2 is alienated take derivative to get the DE of "order 1".