User:Egm6321.f12.team7.parikh.a/R2.3-2

=R*2.3=

Problem 3: Demonstrate affine relationship
Report problem 2.3 from.

Given: A class of N1-ODE
A particular class of nonlinear, 1st order ordinary differential equations can be expressed as

Find: Show affineness
Show that ($$) is affine with respect to $$y'$$. Also explain why ($$) is a non-linear, 1st order, ordinary differential equation. Finally, give an example of a more general nonlinear, 1st order, ordinary differential equation.

Solution: It is affine with respect to y'

 * {| style="width:100%" border="0"

This solution was prepared without referring to previous solutions.
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * }

An affine function or operator is similar to a linear operator, except it is shifted by a constant. Therefore, with respect to $$y'$$, $$M(x,y)$$ can be considered a constant. For a function to be linear, the following must be true.

The linear part of ($$) is given by

This implies that

Therefore, if we substitute $$y' = \alpha u + \beta v$$

Since $$f(\alpha u + \beta v) = \alpha f(u) + \beta f(v)$$, ($$) is linear with respect to $$y'$$ and adding in the constant (with respect to $$y'$$) $$M(x,y)$$ makes ($$) affine with respect to $$y'$$.

However, ($$) is not linear with respect to $$y$$ because the coefficient of $$y'$$ is dependent on $$y$$. Powers greater than one of the dependent variable, or dependent variables multiplied by their derivatives implies nonlinearities. The equation is an ordinary differential equation, as apposed to a partial differential equation, since the dependent variable, $$y$$, is only dependent on one independent variable, $$x$$, and all the derivatives are total derivatives as apposed to partial derivatives. Finally, the equation is 1st order because the highest derivative is the first derivative of the dependent variable.

A more general form of a nonlinear, 1st order, differential equation is given by

where $$y', y$$ and $$x$$ are related nonlinearly. An example is