User:Oh Isaac/R2-7,11

Problem 7: Verify Solutions
Report problem 2.7 from Pea1.f12.sec10.

Given: A Equation
A particular equation:

Find: Show the Process of Verification
Verify that ($$) is the solution of the N1-ODE

from Pea1.f12.sec8.

Solution
According to the elements in ($$), the following calculation is needed.

Substitute the counterparts in the left part of ($$) with ($$) and ($$), we get:

The left part of ($$) equals to its right part, so ($$) is indeed the solution of ($$). (Prove end)

=R*2.11=

Problem 11: Solve The General L1-ODE-VC
Report problem 2.7 from Pea1.f12.sec11.

Given: General L1-ODE-VC
As to a general L1-ODE-VC:

Find: Solution and Proof
1.Solve $$, when

1.

2.

2.Find $$y(x)$$ in terms of$$ a_1(x), a_0(x), b(x)$$.

3.Solve $$, when

1.

2.

Subproblem 1
For general non-homogenous L1-ODE-VC

If $$P(x)\neq 0$$ then $$\forall x$$

Check condition

If the condition is accepted, then

Condition 1
According to the condition $$, $$yields to

We have

Check condition,

The condition is accepted. So we get

Condition 2
According to the condition $$, $$yields to

We have

Check the condition,

The condition is accepted. So we get

Subproblem 2
For $$,

Then we get: