User:Egm6321.f12.team2.van wyk/Report2

=Problem 2.3=

Find
Show that, 2.3.1 is affine in $$y'$$, that it is in general an N1-ODE, but is not the most general N1-ODE. Provide an example of a more general N1-ODE.

Solution
Removing $$M(x,y)$$, we have Since equation 2.3.2 is linear with respect to $$y'$$, and $$G(y',y,x)$$ does not pass through the origin; adding $$M(x,y)$$ makes $$G(y',y,x)$$ nonlinear with respect to $$y'$$, but rather affine with repect to $$y'$$. A more general N1-ODE is of the form: Where $$F(y')$$ is some function of $$y'$$. For example, Which yields,

=Problem 2.4=

Given
The following two equations:

Find
Show that 2.4.1 and 2.4.2 are linearly independent. That is, for any given $$ \alpha \in \mathbb R$$, show that $$\exists \; \hat x$$ such that $$ y_H^1(\hat x) \ne \alpha y_H^2(\hat x)$$

Solution
If $$\hat x = 0$$ then, Therefore, $$y_H^1$$ and $$y_H^2$$ are linearly independent.

=Problem 2.5=

Given
The following function,

Find
The following function, and show that it is a N1-ODE.

Solution
Clearly, the coefficients $$M(x,y),N(x,y)$$ are nonlinear varying coefficients which makes 2.5.6 nonlinear. Also, the highest derivative in this equation is 1, therefore it is of first order. Finally, $$x$$ is the only independent variable in this equation which makes 2.5.6 an ODE. Overall, it is therefore a N1-ODE.