User:Cloudwu890202/Report 6.3

=R*6.3=

Problem 3: Show the Equivalance
Report problem 6.3 from lecture notes section 31-5.

Given: Trial solution in Method 2 and the combined trial solutions in Method 1
One general Equation of Euler Ln-ODE-VC shown as below

which can be also expressed as a compact form below

Two methods for sloving Euler Ln-ODE-VC Method 1: Stage 1: transformation of variables

Stage 2: Trial solution

Method 2: Trial solution

Find: show the Equivalance
show that the trial solution in Method 2 is equivalent ot the combined trial solutions in Method 1

Solution: work with Method 1 and Method 2

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Stage 1
first, we know that $$x=e^t$$, $$\frac{dt}{dx}=e^{-t}$$, thus the general form of Ln-ODE-VC can be written as below

Stage 2
$$\cdot\cdot\cdot$$

Thus, the Ln-ODE-VC can be written as below

To make it more compact, I eliminating$$e^{rt}$$ which is independent with i. thus it can be rewritten as

Method 2
$$\cdot\cdot\cdot$$

Thus, Ln-ODE-VC can be rewritten as

To make it more compact, I eliminating the $$ x^r$$ which is independant with the i.

Apparently, Eq.(3.12) which is generated by Method 1 is equivalant with Eq.(3.19) which is generated by Method 2