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Solving differential equations

  1. Apr 22, 2008 #1
    [SOLVED] Solving differential equations

    1. The problem statement, all variables and given/known data

    Find the solution:
    dy/dx = (1+y^2)/(1+x^2)
    hint: there is a formula tan(a + b) that will simplify your answer.
    y(0)=1
    2. Relevant equations



    3. The attempt at a solution

    I seperated the variables and integrated ending up with

    arctan(y)=arctan(x)+c
    I do not see how to take this to the answer below. I understand how to use the formula they gave me in the hint but I do not see how it applies here...

    the answer is
    y=(1+x)/(1-x) this problem is from my exam review we have the solutions but not how to get them...
     
  2. jcsd
  3. Apr 22, 2008 #2

    exk

    User Avatar

    tan(arctan(y))=y

    take the tangent of both sides and then the formula for tan(a+b) goes to work since you have tan(arctan(x)+c), which is also (tan(a)+tan(b))/(1-tan(a)tan(b)) so you get (x+tan(c))/(1-x*tan(c)).

    y(0)=1 --> tan(c)=1 --> c= arctan(1)

    and you get

    y=(x+tan(c))/(1-x*tan(c)) --> (x+1)/(1-x)
     
  4. Apr 22, 2008 #3
    thanks a lot I was thinking you had to take the tan of both sides but I wasn't sure you cleared it up nicely.
     
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