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Differential equation problem: Solve dy/dx = (y^2 - 1)/(x^2 - 1), y(2) = 2

  • #1

Homework Statement:

Solve dy/dx = (y^2 - 1)/(x^2 - 1), y(2) = 2

Relevant Equations:

dy/dx = (y^2 - 1)/(x^2 - 1), y(2) = 2
This is my attempt:
[tex]\frac{dy}{dx} = \frac{y^2 - 1}{x^2 - 1}
\\ \int \frac{dy}{y^2 - 1} = \int \frac{dx}{x^2 - 1}
\\ \ln \left| \frac{y-1}{y+1} \right| + C_1 = \ln \left| \frac{x-1}{x+1} \right| + C_2
\\ \ln \left| \frac{y-1}{y+1} \right| = \ln \left| \frac{x-1}{x+1} \right| + C [/tex]

Since y(2) = 2,
[tex]\ln \left| \frac{1}{3} \right| = \ln \left| \frac{1}{3} \right| + C
\\ \therefore C = 0[/tex]

So,
[tex]\ln \left| \frac{y-1}{y+1} \right| = \ln \left| \frac{x-1}{x+1} \right|
\\ \left| \frac{y-1}{y+1} \right| = \left| \frac{x-1}{x+1} \right|
\\ \frac{y-1}{y+1} = \pm \frac{x-1}{x+1}
\\y = x, y = \frac{1}{x}[/tex]

Is this ok, or did I make any mistake?
 
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Answers and Replies

  • #2
Orodruin
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Did you try inserting your solutions in the original ODE and boundary condition?
 
  • #3
Did you try inserting your solutions in the original ODE and boundary condition?
Both y = x and y = 1/x satisfy the original ODE, but the boundary condition works for y = x only.
 
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  • #4
Both y = x and y = 1/x satisfy the original ODE, but the boundary condition works for y = x only.
Is that it? Is that why y = x is the only solution?
 
  • #5
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Yes.
 
  • #6
Both y = x and y = 1/x satisfy the original ODE, but the boundary condition works for y = x only.
Is that it? Is that why y = x is the only solution?
If the boundary condition was ##y(1) = 1##, both ## y = x ## and ## y = \frac{1}{x} ## would be correct solutions, right?
 
  • #7
Orodruin
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If the boundary condition was ##y(1) = 1##, both ## y = x ## and ## y = \frac{1}{x} ## would be correct solutions, right?
The differential equation is not well defined in (x,y) = (1,1) as you have an expression of the form 0/0 for dy/dx.
 
  • #8
The differential equation is not well defined in (x,y) = (1,1) as you have an expression of the form 0/0 for dy/dx.
Oh yes. How did I not notice this? :headbang:
 
  • #9
WWGD
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Murshid, did you check the other possibility of y=-x (from your first post) ?
 
  • #10
Murshid, did you check the other possibility of y=-x (from your first post) ?
But my first post mentions only y=x and y=1/x
 
  • #11
WWGD
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But my first post mentions only y=x and y=1/x
I understand , but notice your solution included a ## +/- ##
 
  • #12
I understand , but notice your solution included a ## +/- ##
Yes, ## \frac{y-1}{y+1} = \frac{x-1}{x+1} ## leads to ## y = x ##, and ## \frac{y-1}{y+1} = - \frac{x-1}{x+1} ## leads to ## y = \frac{1}{x} ##
 
  • #13
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Ok, it is not a solution to the original, I just wanted to ask if you had tested it.
 
  • #14
Ok, it is not a solution to the original, I just wanted to ask if you had tested it.
Should I? I don't get this. Why should I test it if it's not a solution to the differential equation? Are you hinting something that I'm missing?
 
  • #15
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Should I? I don't get this. Why should I test it if it's not a solution to the differential equation? Are you hinting something that I'm missing?
My bad, i got confused by the +/- sign. Please disregard.
 

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