Given the system of equations find v_3,v_4. If you only know the value of v_1,v_2, p_0+p=p_1, p_1+p=p_2, p_0+2p=p_3, p_3+p=p_4, p_1 v_1=p_2 v_2=p_3 v_3=p_4 v_4 Came to the equations when solving a complex physics task.

Rosemary Chase

Rosemary Chase

Answered question

2022-11-07

Given the system of equations find v 3 , v 4 . If you only know the value of v 1 , v 2
p 0 + p = p 1
p 1 + p = p 2
p 0 + 2 p = p 3
p 3 + p = p 4
p 1 v 1 = p 2 v 2 = p 3 v 3 = p 4 v 4
Came to the equations when solving a complex physics task.

Answer & Explanation

brulotfao

brulotfao

Beginner2022-11-08Added 16 answers

The first four linear equations have the solution
p = p 4 p 3 , p 0 = 3 p 3 2 p 4 , p 1 = 2 p 3 p 4 , p 2 = p 3
with free parameters p 3 , p 4 . This follows just by adding and subtracting the equations. We can substitute this result to the system of polynomial equations
p 1 v 1 p 2 v 2 = 0 , p 2 v 2 p 3 v 3 = 0 , p 3 v 3 p 4 v 4 = 0.
Assume first that p 3 , p 4 0. Then it follows that
v 3 = v 2 , v 4 = p 3 v 2 p 4 , 2 p 3 v 1 p 3 v 2 p 4 v 1 = 0 ,
and we are done. For the subcases p 3 = 0 or p 4 = 0 it works similarly.

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