An ODE (Ordinary Differential Equation) of order n becomes a relation: F ( x ,...
Lovellss
Answered question
2022-06-22
An ODE (Ordinary Differential Equation) of order n becomes a relation:
Then defines an ODE of order one. In "basic standard texts", for purposes of simplicity, is assumed that some ODE of first order can take the form:
for certain suitable f. Here is my "silly" question: What if that assumpion is not possible?
For example how I can deal with equations of the form:
I appreciate any reference. Thanks in advance for your comments!
Then defines an ODE of order one. In "basic standard texts", for purposes of simplicity, is assumed that some ODE of first order can take the form:
for certain suitable f. Here is my "silly" question: What if that assumpion is not possible?
For example how I can deal with equations of the form:
I appreciate any reference. Thanks in advance for your comments!
Answer & Explanation
Aaron Everett
Beginner2022-06-23Added 18 answers
You can just simply take another derivative to get the equation
which can be transformed into an explicit ODE if is invertible.
In another way, this same condition says that if is invertible and continuous, then by the implicit function theorem the original implicit equation has an explicit solution
Usually, if y is scalar, the points where form a surface and thus the set of regular points is dense, so the ability to resolve into an explicit equation is a stable property.
If y is a vector and F a system of equations of equal dimension, then the rank of is a stable property. One can only resolve it into an explicit ODE if the rank is full. All other cases, excluding some more exotic degeneracies, lead to differential-algebraic equations, DAE.
An ODE has a full vector field on the state space. A DAE only defines direction vectors on a part of the state space, and then usually multiple directions per point. It becomes non-trivial to select directions so that integral curves, i.e., solutions to all defining equations, result.
which can be transformed into an explicit ODE if is invertible.
In another way, this same condition says that if is invertible and continuous, then by the implicit function theorem the original implicit equation has an explicit solution
Usually, if y is scalar, the points where form a surface and thus the set of regular points is dense, so the ability to resolve into an explicit equation is a stable property.
If y is a vector and F a system of equations of equal dimension, then the rank of is a stable property. One can only resolve it into an explicit ODE if the rank is full. All other cases, excluding some more exotic degeneracies, lead to differential-algebraic equations, DAE.
An ODE has a full vector field on the state space. A DAE only defines direction vectors on a part of the state space, and then usually multiple directions per point. It becomes non-trivial to select directions so that integral curves, i.e., solutions to all defining equations, result.
