Differential equations in function. Equations (1) : xy′+(1-x)y=1 let z=xy+1

Darius Miles 2022-09-22 Answered
Differential equations in function
Equations (1): x y + ( 1 x ) y = 1
determine and solve the differential equation (2) whose general solution is the function z .
determine the general solution of (1)
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Answers (2)

soporoseun
Answered 2022-09-23 Author has 8 answers
Step 1
Compute z = y + x y = y + 1 ( 1 x ) y = z
So the general solution of the above equation (call it eqn (2)) is
z ( x ) = z ( 0 ) e x = e x
Step 2
because z ( 0 ) = 1. Solving then for y:
y ( x ) = z 1 x = e x 1 x
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Gillian Cooper
Answered 2022-09-24 Author has 3 answers
Step 1
If z = x y + 1 then
d z d x = d d x ( x y + 1 ) = y + x d y d x
Step 2
Now:
x d y d x + y x y = 1 x d y d x + y = 1 + x y d z d x = z
Now you solve this equation for z and then change back the variables to x and y at the end.
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