User:Egm6321.f11.team2.rho/HW3

=R*3.4-Construct a class of N1-ODEs=

Given
A class of N1-ODES of the form is as follows:

Euler Integrating factor method (IFM) is written as

For (3.4.2) to be exact, 2nd exactness condition is applied to find h and (3.4.2) becomes

To solve (3.4.3), 2 cases is considered.

In the cases, the second case is defined as follows:

Case 2:suppose h(x,y), thus h is a function of y only, and then (3.4.3) becomes

Find
Construct a class of N1-ODEs,which is the counterpart of (3.4.1),and satisfies the condition (3.4.4) that an integrating factor h(y) can be found to render it exact.

Solution
Based on (3.4.4), consider:

Thus:

If  $$\displaystyle k_1(y)=d_1 $$ (constant), then (3.4.4) is satisfied as follows:

=R*3.16- Completion of Second Choice=

Given
From (3.16.1) and (3.16.2), we get

For the first choice of the solution, assume that

The process to find $$\displaystyle \phi(x,y,p)$$ is as follows:

Find
Finish the second choice assuming that

Solution
(3.16.10) can be rewritten as follows:

And then the both sides of (3.16.11) are integrated as follows:

The above integration results in

The result is exactly same with (3.16.5).

Thus, the next process and resulted $$\displaystyle \phi(x,y,p)$$ would be same with the above given functions from (3.16.6) to (3.16.9).

=R3.10 Free Vibration of Coupled Pendulums =

Given
The matrix foam of linear time-variant systems is

The solution of generalized SC-L1-ODE-CC is

Coupled pendulums system is shown as follows:

Pendulums:

{| style="width:100%" border="0" $$\displaystyle a=0.3,l=1,k=0.2 $$
 * style="width:95%" |
 * style="width:95%" |

$$\displaystyle m_{1}g=3,m_{2}g=6,g=10$$

No applied forces:

{| style="width:100%" border="0" $$\displaystyle u_{1}=u_{2}=0$$
 * style="width:95%" |
 * style="width:95%" |

Initial conditions:

{| style="width:100%" border="0" $$\displaystyle \theta_{1}(0)=0,\dot\theta_{1}(0)=-2 $$
 * style="width:95%" |
 * style="width:95%" |

$$\displaystyle \theta_{2}(0)=0,\dot\theta_{2}(0)=1$$

Find
1. Use matlab's ode45 command to integrate the system, (3.10.3) and (3.10.4), in matrix form of (3.10.1) for $$\displaystyle t\in[0,7] $$

2. Use (3.10.2) to find the solution at the same time stations as in Q1.

3. Plot $$\displaystyle \theta_{1}(t) $$ and $$\displaystyle \theta_{2}(t) $$ from Q1 and Q2.

Solution
Solved on my own.

(3.10.3) and (3.10.4) are reordered as follows:

Finally we obtain the following matrix:

Mathlab codes and plot from Q1

Mathlab codes and plot from Q2