User:Egm6321.f11.team4.shin.js/HW2

= Problem 2.3 = From [[media:Pea1.f11.mtg7.djvu|Mtg 7-6]]

Given
The Eq. 2 in the class note [[media:Pea1.f11.mtg7.djvu|7-6]]. The Eq. 1 in the class note [[media:Pea1.f11.mtg7.djvu|7-6]].

Find
Show that the Eq. (3.1) is linear in $$\displaystyle y^{\prime}$$, and that the Eq. (3.1) is in general an N1-ODE. But the Eq. (3-1) is not the most general N1-ODE as represented by the Eq. (3.2). Give an example of a more general N1-ODE.

Show that the Eq. (3.1) is linear in the first derivative of y.
Divide the Eq. (3.1) by $$\displaystyle N(x,y) $$.

Then, the following is true. - The Eq. (3.5) proves that the Eq. (3.1) is linear in $$\displaystyle y^{\prime} $$.

Show that the Eq. (3.1) is N1-ODE.
- The Eq. (3.1) is the 1st order differential equation since the highest order of derivative is 1. - The Eq. (3.1) is Ordinary Differential equation because the differential equation includes one dependent variable and its derivatives.

To see whether the Eq. (3.1) is the linear differential equation, the superposition requirement on the Eq. (3.4) is checked. In other words, the Eq. (3.5) has to be true for the equation to be linear.

Since the RHS of the Eq. (3.6) is not equal to the RHS of the Eq. (3.7), the Eq. (3.1) does not satisfy the superposition condition.

- Since the Eq. (3.1) does not satisfy the superposition, the Eq. (3.1) is nonlinear differential equation.

The Eq. (3.1) is not the most general N1-ODE.
The Eq. (3.1) becomes the 1st order linear ordinary differential equation in particular case when the functions $$\displaystyle M(x,y)$$, $$\displaystyle N(x,y)$$ are in the form of the following.

Since the RHS of the Eq. (3.10) is equal to the RHS of the Eq. (3.11), the Eq. (3.1) satisfies the superposition condition in this particular case. - Therefore, the Eq. (3.1) is not the most general 1st order nonlinear differential equation.

Give an example of a more general N1-ODE.
An example of a general the 1st order nonlinear ordinary differential equation which cannot be converted into the particular form in the Eq. (3.1) is the following. where The Eq. (3.13) can be expressed more generally as following.

where $$\displaystyle F \left( y^{\prime} \right) $$ is any function of $$\displaystyle y^{\prime} $$ (nonlinear in general). If the function $$\displaystyle F(\cdot) $$ has no explicit inverse $$\displaystyle F^{-1}(\cdot) $$, then the Eq. (3.15) cannot be converted into the Eq. (3.1).

Author
Contributed by Shin

=''' Problem 2.4 - Show that the Eq. (3) in page 8-1 is a N1-ODE. '''= From [[media:Pea1.f11.mtg8.djvu|Mtg 8-1]]

Given
The Eq. 3 in the class note [[media:Pea1.f11.mtg8.djvu|8-1]].

Find
Show the Eq. (4.1) is non-linear 1st order ODE.

Solution
- The Eq. (4.1) is 1st order equation since the largest number of differentiation is 1 as shown in the Eq. (4.2). - The Eq. (4.1) is Ordinary Differential equation because the differential equation includes one dependent variable and its derivatives. To see whether the Eq. (4.1) is the linear differential equation, the superposition requirement on the Eq. (4.3) is checked. In other words, the Eq. (4.4) has to be true.

Since the RHS of the Eq. (4.5) is not equal to the RHS of the Eq. (4.6), the Eq. (4.1) does not satisfy the superposition condition. - Since the Eq. (4.1) does not satisfy the superposition, the Eq. (4.1) is nonlinear differential equation.

Author
Contributed by Shin

=''' Problem 2.5 - Explain the Eq. (4) in p.8-1 and the Eq. (1) in p.8-2 '''= From [[media:Pea1.f11.mtg8.djvu|Mtg 8-2]]

Given
The Eq. (4) in the class note [[media:Pea1.f11.mtg8.djvu|8-1]] and the Eq. (1) in the class note [[media:Pea1.f11.mtg8.djvu|8-2]].

The Eq. (1) in the class note [[media:Pea1.f11.mtg8.djvu|7-6]] and the Eq. (2) in the class note [[media:Pea1.f11.mtg8.djvu|7-6]].

Find
Explain the Eq. (5.1) satifies the the Eq. (5.3) but can be converted to the Eq. (5.4) with the definition in the Eq. (5.2).

Solution
Move the first term in the LHS of the Eq. (5.1) to the LHS. Take the cube root on both side. Move back the term in the RHS to the LHS. Then, the Eq. (5.7) can be converted into the form of the Eq. (5.4) as shown below. when

Author
Contributed by Shin