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Legendre Polynomials

17. The first three Legendre polynomials are P0(x) =1 , P1(x) = x and P2(x)=1/2(3x^2-1). If x = cos θ, then P0(cos θ)=1 and P1(cos θ)= cos θ. Show that p2(cos θ)= 1/4(3 cos 2θ +1).

Book:- Differential Equations, by Dennis G Zill, page ,number 17.


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a) Find the equivalent resistance of the circuit.

600 and 400 resistances are in parallel and their equivalent is
given by
1. R1 = 600*400/(600 + 400) = 240 

2. Now this resistance R1 in series with 560  resistor hence their equivalent is given by
R2 = 240 + 560 = 800 

3. This R2 = 800 is in parallel with given 800 and hence their equivalent resistance or the resistance of the whole circuit is given as

R = 800*800/(800 + 800) = 400

b) Find the current through the battery.
As the equivalent resistance of the circuit is 400 the current through the Battery (considering internal resistance of the battery to be negligible) is given by
I = V/R = 12/400 = 0.03 A = 30 mA.

c) Find the voltage across the 800 ohm resistor.

As the ends of 800 resistor are directly connected to the battery terminals the voltage across it is same as the voltage across the terminals of the battery and is 12 Volts.

c) Find the current in the 560 ohm resistor.

The ends of the combination of 600, 400 and 560  are connected to the battery terminals, the voltage across it is 12 volts and the current through the combination is the ratio of voltage and the equivalent resistance of the combination R2 = 800 and hence current through the 560 resistance (same as the combination) is

I = 12/800 = 0.015 A = 15 mA.

d) Find the voltage across the 400 ohm resistor.

Voltage drop across 560  resistor will be (I*R) = 0.015*560 = 8.4 V

Hence the voltage drop across the parallel ...

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Legendre polynomials are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.