User:EGM6321.f12.team4.Rui.C/Homework4 R4.4(3)

Solution (3)
The form of $$\phi(x,y,p)$$ can be written as:

So a L1-ODE-VC can be solved to get y(x):

It is of the form:

where

So the first exactness condition holds.

For the second exactness condition, since

The second exactness does not hold for the original L1-ODE-VC. IFM will be used. Let:

Then:

A PDE for $$h(x,y)$$ should be solved in order to derive the first integral:

Let $$h=h(x)$$. Then the PDE can be simplified as:

The integration is difficult to be computed analytically. We write $$h(x)$$ as

Then the ODE can be written as:

Substituting

into (4.4.3.14) to replace the second h(x), we have

which is the final expression for y and here $$h(x)$$ is denoted by (4.4.3.14).

Author and References

 * Solved and Typed by -- Jinchao Lu and Rui Che