User:Egm6321.f11.team4.shin.js/HW4

= Problem R*4.4 - Derivation of a form of exact L2-ODE-VC = From [[media:Pea1.f11.mtg13.djvu|Mtg 13-5]]

Given
The equation (2),(3) in the class note [[media:Pea1.f11.mtg13.djvu|21-4]]. The equation (1) in the class note [[media:Pea1.f11.mtg13.djvu|21-5]].

Find
Show that (2),(3) in the class note [[media:Pea1.f11.mtg13.djvu|21-4]] leads to the equation (1) in the class note [[media:Pea1.f11.mtg13.djvu|21-5]].

Solution
- Solved on our own The first exactness condition for L2 ODE VC: where $$\displaystyle p = y^{\prime} $$. By comparing the Eq. (4.4) and the Eq. (4.1), the following is obtained. Integrate the Eq. (4.5), By differentiating the Eq. (4.6) with respect to x and y, the following is obtained. Substituting the Eq. (4.7) into the Eq. (4.4) yields the following. From the last term of the Eq. (4.8), the following is true. Integrate the Eq. (4.9) with respect to $$\displaystyle x $$. Substitute the Eq. (4.10) into the Eq. (4.6). Take the partial derivative of the Eq. (4.11) with respect to $$\displaystyle y $$. Also, the following can be proven to be true using the problem statement. Hence, by comparing the Eq. (4.12) and the Eq. (4.13), the following can be concluded. Therefore, the following is justified.

Author
Contributed by Shin