User:Egm6321.f12.team4.lu/Homework4 R4.4

Problem 4.4
sec18-2 sec21-6

Statement
L2-ODE-VC

Find
1.Show equation 4.5.1 is exact

2.Find Φ

3.Slove for $$y(x)$$

1.Show equation 4.4.1 is exact
Let

We obtain

It satisfies the 1st exactness condition

Recall the 2nd exactness condition for L2-ODE-VC:

We substitute all equations above into (4.4.3) and (4.4.4), we obtain

It satisfies the 2nd exactness condition

It is proved

End

2.Find Φ
We know (4.4.1) is exact,so we obtain

We obtain

Finally, we obtain

End

3.Solve for $$y(x)$$
Recall that

From (4.4.5) and (4.4.6), we obtain

Rearrange, we have

Use integration factor

Finally, we obtain

End

Author and References

 * Solved and Typed by -- Jinchao Lu