User:Egm6321.f12.team2.mohan/Report2

=Problem 2.7=

Verify That
is indeed the solution for the N1-ODE,

Solution
Move individual variables to their respective sides of the equation,

Take the integral of both sides,

Resulting in,

And hence,

=Problem 2.8=

Solution
=Problem 2.9=

Find
$$ h $$ which is the Euler Integrating Factor

Solution
Considering $$h_y$$ as the integration factor, and replacing $$\frac{1}{M}(N_x-M_y)$$ with $$m(y)$$,

Solving the integrals, leads to:

And then solving for $$h$$, we get:

Which can be further simplified as general statement to:

=Problem 2.10=

Show That
and

Solution
Rewrite (2.10.1) as the following:

where, $$M(x,y)$$ and $$N(x,y)$$ are the variables in an ODE. In this case $$N(x,y)=1$$ Equation (2.10.4) is not EXACT, because $$M_x = x^{-1} \neq N_y = 0$$ To make them EXACT, we multiply (2.10.4) by $$h(x)=x$$, resulting in,

Now, $$M_y = 1$$ and $$N_x = 1$$, so $$M_y=N_x$$ making the ODE EXACT and the integrating factor be $$h(x)=x$$ We can now integrate $$M(x,y)$$ and $$N(x,y)$$ from (2.10.4) individually, Resulting in,

And solving for $$y$$ in (2.10.8) we get: