dx/x(y^2-z^2)=dy/-y(z^2+x^2)=dz/(z(x^2+y^2)
Answers
Answer:
The general solution to the system of differential equations is:
|x| = |y| / sqrt(x^2+z^2)
|y| = |z| * sqrt(x^2+y^2)
Step-by-step explanation:
To solve this system of differential equations, we can use the method of separation of variables.
Starting with the first equation:
Dx/x(y^2-z^2) = dy/-y(z^2+x^2)
We can cross-multiply and integrate both sides with respect to their respective variables:
∫ Dx/x = ∫ dy/y(z^2+x^2)
ln|x| = -ln|y| - arctan(z/x) + C1
where C1 is the constant of integration.
Similarly, for the second equation:
dy/-y(z^2+x^2) = dz/(z(x^2+y^2))
Again, cross-multiply and integrate:
∫ dy/y = ∫ dz/z(x^2+y^2)
ln|y| = ln|z| - arctan(x/y) + C2
where C2 is the constant of integration.
We now have two equations involving the constants of integration C1 and C2. To solve for these constants, we can use the initial conditions of the system.
For example, if we are given that the system passes through the point (1,1,1), then we can substitute x=1, y=1, and z=1 into the equations above to obtain two equations involving only C1 and C2:
ln|1| = -ln|1| - arctan(1/1) + C1
ln|1| = ln|1| - arctan(1/1) + C2
Simplifying, we get:
C1 = 0
C2 = 0
Now, substituting these values of C1 and C2 back into the equations above, we get:
ln|x| = -ln|y| - arctan(z/x)
ln|y| = ln|z| - arctan(x/y)
Exponentiating both sides of each equation, we get:
|x| = e^(-ln|y|) * e^(-arctan(z/x))
|y| = e^(ln|z|) * e^(-arctan(x/y))
Simplifying further, we get:
|x| = |y| / sqrt(x^2+z^2)
|y| = |z| * sqrt(x^2+y^2)
Therefore, the general solution to the system of differential equations is:
|x| = |y| / sqrt(x^2+z^2)
|y| = |z| * sqrt(x^2+y^2)
Note that the absolute value signs are included to account for all possible signs of x, y, and z.
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