User:Cloudwu890202/Report 6.6

=R*6.6=

Problem 6: Special IFM to solve nonhomogeneous L2-ODE-VC con't
Report problem 6.6 from lecture notes section 33-1.

Given: Nonhomogeneous L2-ODE-CC cont'd
One general Equation of Euler L2-ODE-VC shown as below

Find: some specific Questions
1. Find the PDEs that govern the integrating factor h(x,y) for the Eq.(6.1) Recall the 2 relations in the 2nd exactness condition for N2-ODEs. solve these PDEs for h(x,y) 2. Trial solution for the integrating factor $$ h(t)= e^{\alpha t}$$, which is similar to the trial solution for the Euler L2-ODE-CC (a homogeneous Ln-ODE-CC), where $$ \alpha$$ is unknown to be determined.

Because of the integrating factor in exponential form, assume the l.h.s. take the form:

Clearly $$\bar a_2=0$$ to avoid having $$y'''$$ when differentiating the r.h.s; an advantage is to reduce the order of the resulting ODE.

2.1. Find $$(\bar a_1,\bar a_0)$$ in terms of $$(a_0,a_1,a_2)$$ 2.2. Find the quadratic eqution for $$\alpha $$ Use the Eq. shown as below 2.3. Reduced-order equation: Eq.(6.3) and Eq.(6.4) lead to

a L1-ODE-CC, thus easily solvable by the IFM

2.4. Use the IFM to solve

2.5. show that

thus $$(\alpha,\beta)$$are roots of the quadratic equation:

which is the same as Eq.(6.6)

2.6. deduce the particular solution $$y_P(t)$$ for general excitation $$f(t)$$

2.7. verify result with table of particular solutions for:

2.8. solve the nonhomogeneous L2-ODE-CC with the following excitation: Gaussian distribution:

for the coefficients $$ (a_0,a_1,a_2) $$, consider two different characteristic equations: 2.8.1.

2.8.2

2.9. for each case in 2.8.1 and 2.8.2, determine the fundamental period of undamped free vibration. Plot the homogeneous soln $$y_H(t)$$ for about 5 periods, the partucular solution $$y_P(t)$$ for the excitation for the same time interval, and the complete solution $$y(t)$$, assuming ero initial conditions.

Solution: solve the problems

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Part 1
Using the IFM to solve the problem, thus the general form of Ln-ODE-VC can be written as below

Because $$ h(t,y) $$ is a function only with respect of y. Thus $$ h(t,y) $$ can be rewritten as $$ h(t) $$ Eq.(6.15) can be rewritten as

which is satiesfied the 1st exactness condition:

where

Then calculate that

Then examine that with the 2nd exactness condition

which can be specified

solve the 2 eqn, then yeild

Thus this problem can be solved by solving Eq.(6.23)

2.1
Given that

Substituting Eq.(6.24) into Eq.(6.23)

Substituting Eq.(6.24) into Eq.(6.3) Because $$ a_2=o$$, thus Eq. (6.26) can be rewritten as

differentiate the right side of the Eq.(6.27)

Solving for $$(\bar a_1,\bar a_0)$$ in terms of $$(a_0,a_1,a_2)$$ as below

2.2
Equate Eq.(6.30) and Eq.(6.31) yeild

which can be rearranged as below

2.3
Thus

2.4
Solve the problem by IFM

By definition

Then the integration of the h(t) can be rewritten as below

Thus

where

Substitute the $$ \bar h(t) B(t)$$ into the Eq.(6.41) yeilds

2.5
Substituting Eq.(6.29)(6.30)(6.31) into the Eq. shown below

Thus

2.6
By Euler L2-ODE-CC, I add the integrating constants $$k_1, k_2$$ to make Eq.(6.42) more general

where

2.7
substitute $$ f(t)=te^{bt}$$ into the Eq.(6.45) yeild

use Wolfram-Alpha integrating Eq.(6.46) yeilds

Which can be rewritten as

2.8.1
Given that

two solution are$$ r_1=-1,r_2=2 $$,in other words,$$ \alpha=-1,\beta=2 $$ Thus, two homogeneous solution for the original function are

As Gaussian distribution

use Wolfram-Alpha integrating Eq.(6.46) yeilds

Thus,

2.8.2
Given that

two solution are$$ r_1=r_2=4 $$,in other words,$$ \alpha=\beta=4 $$ Thus, two homogeneous solution for the original function are

As Gaussian distribution

use Wolfram-Alpha integrating Eq.(6.46) yeilds

Thus