User:Egm6321.f11.team2.rho/HW4

=R*4.4- Derivation of a class of exact L2-ODE-VC=

Given
The form of general N2-ODE is

The (4.4.1) is considered as follows:

A generated class of exact L2-ODE-VC is

Find
Show that (4.4.2) and (4.4.3) leads to (4.4.4).

Solution
From (4.4.1) and (4.4.2), we can obtain

Integrating the both sides of (4.4.5), the following is obtained.

Using (4.4.6),

whrere

$$ \displaystyle \begin{align} &p = y'(x) \\ &\phi_x = P'(x)p + \frac{\partial k(x,y)}{\partial x} \\ &\phi_y = \frac{\partial k(x,y)}{\partial y} \end{align} $$

Integrating the term of R(x)y in (4.4.7),

With (4.4.8), (4.4.6) can be rewritten as follows:

To find k(y), $$\displaystyle \phi_y $$ is found and compared to Q(x) because $$\displaystyle \phi_y = Q(x) $$ in (4.4.7).

Frrom (4.4.10), we can find the following:

Therefore, k(y) is verified with constant k and (4.4.9) is resulted in

$$ \displaystyle \phi(x,y,p) = P(x)p + T(x)y + k $$