- #1
Apothem
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Problem:
y'=((x-1)/(x^2))*(y^2) , y(1)=1 . Find solutions satisfying the initial condition, and determine the intervals where they exist and where they are unique.
Attempt at solution:
Let f(x,y)=((x-1)/(x^2))*(y^2), which is continuous near any (x0,y0) provided x0≠0 so a solution with y(x0)=y0 is guaranteed to exist when x0≠0
The partial derivative with respect to y of f(x,y) is 2y*((x-1)/(x^2)) which is continuous near any (x0,y0) provided x0≠0 so a solution with y(x0)=y0 is unique when x0≠0 (By Picard's Theorem)
From this the interval in which the solutions are unique is x∈(-∞,0)∪(0,∞).
Solving the differential equation and using the initial condition y(1)=1 we see that y(x)=x/(ln|x|+1). The interval of existence is 1/e < x < ∞
Is this right, or...?
Thanks for any help!
y'=((x-1)/(x^2))*(y^2) , y(1)=1 . Find solutions satisfying the initial condition, and determine the intervals where they exist and where they are unique.
Attempt at solution:
Let f(x,y)=((x-1)/(x^2))*(y^2), which is continuous near any (x0,y0) provided x0≠0 so a solution with y(x0)=y0 is guaranteed to exist when x0≠0
The partial derivative with respect to y of f(x,y) is 2y*((x-1)/(x^2)) which is continuous near any (x0,y0) provided x0≠0 so a solution with y(x0)=y0 is unique when x0≠0 (By Picard's Theorem)
From this the interval in which the solutions are unique is x∈(-∞,0)∪(0,∞).
Solving the differential equation and using the initial condition y(1)=1 we see that y(x)=x/(ln|x|+1). The interval of existence is 1/e < x < ∞
Is this right, or...?
Thanks for any help!