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=R*6.3 - Show that Method 1 and Method 2 for solving Euler Ln-ODE-VC are the same=

Given
Two methods for solving Euler Ln-ODE-VC: Method 1: Stage 1: Transformation of variables Stage 2: Method 2: Trial Solution

Find
Show that Method 1 is equivalent to Method 2.

Solution
Substituting ($$) into ($$) yields ($$):

Given
The following characteristic equation,

the following Euler L2-ODE-VC,

the first homogeneous solution,

and the following Euler L2-ODE-CC, .

1.1
Find $$a_2,a_1,a_0$$ such that ($$) is indeed the characteristic equation of ($$)

1.2
Find the complete solution to ($$),

1.3
Find the second homogenous solution, $$y_2(x)$$

1.4
Find $$a_2,a_1,a_0$$ such that ($$) is indeed the characteristic equation of ($$)

1.5
Find the complete solution to ($$),

1.6
Find the second homogenous solution, $$y_2(x)$$

1.1
Rewriting the characteristic equation yields, Since ($$) is an Euler L2-ODE-VC, we must transform the variable with $$x=e^t$$ which implies that the characteristic equation corresponds to, Using the relations previously established in lecture, we know that from (2) p.30-7, and from (5) p.30-6, .

Rearranging ($$) and ($$) and using $$x=e^t$$ yields,

Substituting ($$) and ($$) into ($$) yields, .

Therefore, $$a_2=1,a_1=-9,a_0=25$$.

1.2
The complete solution is $$y(x)=c(x)y_1(x)$$ From the previously developed relations in lecture, we know the following:

From (2) p.34-5,

From (1) p.34-5,

From (4) p. 34-4,

Since ($$) is homogeneous, $$f(x)=0$$. Also, $$a_1(x)=-9x$$. Therefore,

Furthermore,

and,

Finally,

1.3
Distributing ($$) yields,

where the second homogeneous solution is

1.4
Since ($$) has constant coefficients, the coefficients are obtained directly from the characteristic equation $$b_2=1,b_1=-10,b_0=25$$

1.5
The previously mentioned relations are used again to obtain the full solution,

Following,

Then,

Finally,

1.6
Distributing ($$) yields,

Therefore, the second homogeneous solution is

Given
The following non-homogeneous Legendre equation L2-ODE-VC,

and its first homogeneous solution, .

Find
The final solution, $$y(x)$$ by variation of parameters

Solution
First, ($$) is rewritten as the following,

The complete solution is $$y(x)=U(x)P_1(x)$$. From the previously developed relations in lecture, we know the following:

From (2) p.34-5,

From (1) p.34-5,

From (4) p. 34-4,

Therefore,

Following,

Next,

Finally,