User:Egm6321.f12.team4.lu/Homework3 R3.2&R3.8

Problem 3.2
sec11-3 sec11-5

Statement
Show that the solution of

in

agrees with the result presented in King 2003 p.512, i.e.,

Use

and

to identify $$A$$, $$y_H(x)$$ and $$y_P(x)$$. Compare your results with those in King 2003 p.512

Solution
King defines the L1-ODE-VC as

integrating factor as

where $$ t $$ is a dummy variable

And equation

is defined with

where $$ A $$ is constant of integration

Substitute (3.2.4) into (3.2.2), and then expand the equation, we obtain

From (3.2.10), we can separate this equation with respect to (3.2.8) and (3.2.9), thus, we obtain,

Compare (3.2.8) with (3.2.11), it shows that

Compare (3.2.9) with (3.2.12), it shows that

End

Author and References

 * Solved and Typed by -- Jinchao Lu

Problem 3.8
sec11-3 sec12-3 sec13-2

Statement
Construct a class of N1-ODEs, which is the counterpart of

and satisfies the condition

that an integrating factor $$ h(y) $$ can be found to render it exact

Solution
To satisfy the above condition, consider

Integrate (3.8.4), (3.8.5) and (3.8.3)

We obtain,

Thus, we can determine a counterpart of (3.8.1)

Where $$a(y)$$,$$b(y)$$,$$c(x)$$ are arbitrary functions.

End

Author and References

 * Solved and Typed by -- Jinchao Lu