User:Egm6321.f12.team4.lu/Homework5 R5.8&R5.9

Problem R*5.8-Show equivalence of two forms of 2nd exactness condition
sec16-5 sec21-7 sec22-5 sec22-6

Statement
Case n=2:N2-ODE

Form2

Form1

or equivalently 2nd exactness condition for N2-ODEs

Show this equivalence

Solution
From sec22-5(1)

Define

From sec21-7(3)

We obtain

Derivation

From Eq(5.8.6)

From Eq(5.8.7)

Put Eq(5.8.5), Eq(5.8.8) and Eq(5.8.9) into Eq(5.8.1), yields

Define

So, Eq(5.8.10) can be transferred to

Since 1 and q are linearly independent, we must have

Finally we have

It is proved. The same to sec16-5(2)

End

Author and References

 * Solved and Typed by -- Jinchao Lu
 * Reviewed by --

Problem R5.9-Use Taylor series to derive
sec64-4 sec64-5 sec64-7

Statement
Use Taylor series at $$x=0$$ (aka Maclaurin series) to derive

and

Solution
First, recall Taylor series

At $$x=0$$, it means that $$a=0$$ in Eq(5.9.1), yields

i) $$f(x)=(1-x)^{-a}$$
Now consider the derivation

We obtain

Substitute Eq(5.9.3) into Eq(5.9.2), we obtain

Recall Pochhammer's symbol

So, let $$n=k$$ Eq(5.9.4) can be transferred to

Recall hybergeometric function

Finally, we obtain

End

ii) $$f(x)=\frac{1}{x}\arctan(1+x)$$
Before I start to solve this problem, Rui Che and I recognize that the equation in the lecture(sec64-7) is totally wrong. So we modify it in our assumption.

We are solving $$f(x)=\frac{1}{x}\arctan(x)$$ in stead of $$f(x)=\frac{1}{x}\arctan(1+x)$$

Recall

We obtain

Now consider the derivation

We obtain

Put Eq(5.9.6) and Eq(5.9.7) into Eq(5.9.2), yields

Let $$ n=2k $$

Rearrange Eq(5.9.9), we obtain

Finally, we obtain

End

Author and References

 * Solved and Typed by -- Jinchao Lu
 * Reviewed by --