User:EGM6321.f12.team7.Marjanovic/report3/3.5

=R*3.5=

Problem 5: Show that a Nonlinear 1st Order ODE meets certain conditions
Based on lecture notes section 13

Given: An N1-ODE
The following N1-ODE

where

has an integrating factor h(x) that can be found to render it exact using:

only if

$$\displaystyle k_1(y) = d_1 $$ (constant)

Find: Support that for the N1-ODE k(y) must be constant
Prove that the above statement is true. Furthermore, show that ($$) includes

as a particular case.

Solution: k(y) is constant and 5.5 is a particular case of 5.1

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Since

$$\displaystyle N(x,y) = \bar b(x,y)c(y) $$

$$\displaystyle M(x,y) = a(x)\bar c(x,y) $$

then, using ($$) and ($$),

Equation ($$) becomes:

Since $$\displaystyle n(x) $$ is defined as only a function of x, $$\displaystyle k_1(y) $$ must be a constant for that to be true.

Now to prove ($$) is a particular case, let us suppose that

$$\displaystyle c(y) = 1 $$

$$\displaystyle k_1(y) = 0 $$

We would then have,

$$\displaystyle \bar b(x,y) := \int^x b(s)ds + 0 = \bar b(x) $$

$$\displaystyle \bar c(x,y) := \int^y 1ds + k_2(x) = y + k_2(x) $$

$$\displaystyle c(y) $$ then becomes

$$\displaystyle c(y) = \frac{\partial (\bar c(x,y))}{\partial y} = 1 $$

Since a(x) is arbitrary, ($$) then reduces down to ($$) or:

$$\displaystyle [a(x)y+k_2(x)] + \bar b(x)y' = 0 $$

Therefore it is a particular case of ($$)