User:EGM6321.f12.team4.Rui.C/Homework6 R6.3

Statement
Trial solution (5)p.30-4 in Method 2 can be written as:

while the combined trial solutions (3)-(4)p.30-4 in Method 1 can be written as:

Solution
In Method 2, trial solution (6.3.0.1) is substituted directly into the following Euler Ln-ODE-VC:

while in Method 1, (6.3.0.2) is first substituted into (6.3.1.1) to replace x with t. Then a trial solution for y with respect to t (6.3.0.3) is substituted into equation (6.3.1.1).

In Method 2, we inspect a particular term: $$a_ix^iy^{(i)}$$ :

In Method 2, we first substitute (6.3.0.2) into the specific term in the left hand side of (6.3.1.1) and recall that:

So:

We then plug (6.3.0.3) into the above expression to replace $$y$$ with $$e^{rt}$$ :

We then substitute (6.3.0.2) back into (6.3.1.5) to replace t in terms of x:

which is the same expression as (6.3.1.2). Thus Method 1 is equivalent to the direct Method 2.

Author and References

 * Solved and Typed by -- Rui Che