Find the equation of the line that passes through the point (0,2) and perpendicular to the line x-y=6

fofopausiomiava 2022-09-06 Answered
Find the equation of the line that passes through the point (0,2) and perpendicular to the line x-y=6
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Answers (1)

Rihanna Blanchard
Answered 2022-09-07 Author has 13 answers
Let,the equation of the line be y=mx+c where, m is the slope and c is the Y intercept.
now,slope of the equation x−y=6 or y=x−6 is 1
We know that for two lines to be mutually perpendicular,their product of slope will have to be −1
So,m*1=−1
or, m=−1
So,the equation becomes, y=−x+c
Now,given the line passes through (0,2)
So,putting the values we get,
2=0+c
So,the equation of the line is, y=−x+2
or, x+y=2
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I have a system of coupled differential equations (an example )in the form of,
x + a x + b x c y = 0
y a y + b y c x = 0
The solution to the above system looks like,
x = A e w 1 t + B e w 2 t + C e w 3 t + D e w 4 t
y = E e w 1 t + F e w 2 t + G e w 3 t + H e w 4 t
the frequencies w 1 , w 2 , w 3 and w 4 are functions of a,b and c My goal is to determine that values of a,b and c such that all w are Real. However, my original system of differential equations has nonlinear terms. I managed to derive the solution but in power expansion form or,
x = i = 0 n a i i ! t i
y = i = 0 n a i i ! t i
I need to determine the frequencies so that I can find the values of a,b and c such that all w are Real. Is it possible to derive w from limited power expansion (say 8th term) of x and y?
My first attempt:
I did a test of my method by taking Fourier Sine Transform of analytical solution which give me the answer in form of f ( w ) g ( w ) where w is the frequency. Then solved the g(w) for w which gives me four answers for w which are the frequencies w 1 , w 2 , w 3 and w 4 . However when I attempted with Fourier Sine Transform of series solutions, the answer is different. This is due to the limited power expansion. Is here a way to improve this?
My second attempt:
I linearize the nonlinear terms of differential equations and used the matrix to calculate Determinant of (A-wI) where A is the matrix of the system of differential equation. I managed to calculate the values of a,b and c but they were incorrect because of linear terms I made.
A last method that I am considering is Monte Carlo. Is it possible to get a frequency equation from limited power expansion of differential equation solution? Any ideas for other methods that I missed?