y′(x)=x+y/z z′(x)=x−y/y

Haiden Meyer 2022-09-28 Answered
I have a system of differential equations which I need to solve and obtain y(x) and z(x). I tried elimination method and got to a point but I don't know what to do after here. Any help would be appreciated Question:
y ( x ) = x + y z
z ( x ) = x y y
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Bestvinajw
Answered 2022-09-29 Author has 15 answers
y ( x ) = x + y z z ( x ) = x y y
y ( x ) z = x + y z ( x ) y = x y
Sum both differential equations:
y z + z y = 2 x
( y z ) = 2 x
y z = x 2 + C
The second DE is:
z ( x ) y = x y
z = x y 1
z = z x x 2 + C 1
That you can solve.

Some details
z = z x x 2 + C 1
You can't integrate both sides the way you did because there is the z function on RHS:
z z x x 2 + C = 1
x 2 + C z z x x 2 + C = x 2 + C
( z x 2 + C ) = 1 x 2 + C
Now you can integrate both sides.
z x 2 + C = d x x 2 + C + C 2
z ( x ) = x 2 + C ( C 2 arctan x x 2 + C )
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more