Lovellss

2022-06-22

An ODE (Ordinary Differential Equation) of order n becomes a relation:

$F(x,y,{y}^{(1)},...,{y}^{(n)})=0$

Then $F(x,y,{y}^{(1)})=0$ 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:

${y}^{(1)}=f(x,y)$

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:

${({y}^{(1)})}^{5}+sen({y}^{(1)})+{e}^{{y}^{(1)}}+x=0$

I appreciate any reference. Thanks in advance for your comments!

$F(x,y,{y}^{(1)},...,{y}^{(n)})=0$

Then $F(x,y,{y}^{(1)})=0$ 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:

${y}^{(1)}=f(x,y)$

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:

${({y}^{(1)})}^{5}+sen({y}^{(1)})+{e}^{{y}^{(1)}}+x=0$

I appreciate any reference. Thanks in advance for your comments!

Aaron Everett

Beginner2022-06-23Added 18 answers

You can just simply take another derivative to get the equation

${\partial}_{x}F+{\partial}_{x}{F}_{{y}_{0}}\xb7{y}^{\prime}+{\partial}_{{y}_{1}}F\xb7{y}^{\u2033}+\dots +{\partial}_{{y}_{n}}F\xb7{y}^{(n+1)}=0$

which can be transformed into an explicit ODE if ${\partial}_{{y}_{n}}F$ is invertible.

In another way, this same condition says that if ${\partial}_{{y}_{n}}F$ is invertible and continuous, then by the implicit function theorem the original implicit equation has an explicit solution

${y}^{(n)}=g(x,y,{y}^{\prime},\dots ,{y}^{(n-1)}).$

Usually, if y is scalar, the points where ${\partial}_{{y}_{n}}F=0$ 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 ${\partial}_{{y}_{n}}F$ 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.

${\partial}_{x}F+{\partial}_{x}{F}_{{y}_{0}}\xb7{y}^{\prime}+{\partial}_{{y}_{1}}F\xb7{y}^{\u2033}+\dots +{\partial}_{{y}_{n}}F\xb7{y}^{(n+1)}=0$

which can be transformed into an explicit ODE if ${\partial}_{{y}_{n}}F$ is invertible.

In another way, this same condition says that if ${\partial}_{{y}_{n}}F$ is invertible and continuous, then by the implicit function theorem the original implicit equation has an explicit solution

${y}^{(n)}=g(x,y,{y}^{\prime},\dots ,{y}^{(n-1)}).$

Usually, if y is scalar, the points where ${\partial}_{{y}_{n}}F=0$ 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 ${\partial}_{{y}_{n}}F$ 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.