User:Egm6321.f11.team2.rho/HW2

=R*2.4- Verification of Nonlinearity in a Particular First-Order ODE=

Find
Verify that (2.4.1) is a N1-ODE.

Solution
[[media:pea1.f11.mtg8.djvu|General nonlinear ODE of order 1 (N1-ODE)]] is

(2.4.1) is a statement of (2.4.2) showing that the highest order is one. Thus, (2.4.1) is a fist order differential equation.

[[media:pea1.f11.mtg4.djvu|Linearty]] has to satisfy the following condition:

[[media:pea1.f11.mtg5.djvu|Nonlinearty ]] is the condition not satisfing the above statement as follows:

(2.4.1) can be rewritten as follows:

And then, the nonlinearity can be checked with the following two functions:

From (2.4.6) and (2.4.7), $$\displaystyle F(\alpha y)\neq\alpha F(y)$$ is found so that (2.4.1) is verified to be a nonlinear equation.

Finally it is concluded that (2.4.1) is a N1-ODE (Nonlinear First Oder Differential Equation).

=R*2.10- Verification of solution for N1-ODE=

Given
where $$\displaystyle M(x,y)=75x^4$$ and $$\displaystyle N(x,y)=cos y$$

Find
Verify that (2.10.1) is the solution for the (2.10.2).

Solution
(2.10.1) would be rewritten as follows:

Differentiating the both sides of (2.10.3), (2.10.3) results in

Thus, (2.10.1) is verified to be the solution for (2.10.2.)

=R*2.16-Solution of L1-ODE-VC=

Given
Integrating factor is

The derivated result in King 2003 p.512 is

Find
Show that (2.16.1) agrees with (2.16.3).

Solution
General L1-ODE-VC is

In King 2003 p.512, the L1-ODE-VC is presented as follows:

From (2.16.3) and (2.164), the following relations can be derived.

From the relations, in King 2003 P512, the integrating factor is shwon as follows:

When (2.16.1) is substituted by (2.16.9), (2.16.1) becomes

$$\displaystyle \Rightarrow y(x)= k_2\exp\left(-\int^x P(t)dt\right)+\exp\left(-\int^x P(t)dt\right)\int^x \exp\left(\int^x P(t)dt\right)b(s)ds$$

Substituting the relation (2.16.8) for (2.16.10), (2.16.10) can be rewritten as follows:

(2.16.10) is the solution presented in King 2003 p.512 identifying A, $$\displaystyle y_H(x)$$ and $$\displaystyle y_p(x)$$ as follows:

Thus, it is verified that (2.16.1) agrees with (2.16.3).