anonymous
  • anonymous
I need to find the partivular solution to the differential equation dy/dx=(1+y)/x Given the initial conditon f(-1)=1
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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bahrom7893
  • bahrom7893
I love diff equations! =) So first of all see if its separable
bahrom7893
  • bahrom7893
it is!
bahrom7893
  • bahrom7893
So separate it: \[dy/dx = (1+y)/x\]

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bahrom7893
  • bahrom7893
Sorry, I keep crashing on here but i will always reply: So multiply both sides by dx and divide both sides by 1+y
bahrom7893
  • bahrom7893
You will have: \[dy/(1+y) = dx/x\]
anonymous
  • anonymous
can't you think of it as dy/(1+y) you're in good hands later.
bahrom7893
  • bahrom7893
Integrate both sides: Left side will be just Ln|1+y| and the right side will be Ln|x|
bahrom7893
  • bahrom7893
So: Ln|1+y| = Ln|x| + C
bahrom7893
  • bahrom7893
Raise e to both sides to get rid of Ln: \[e^{Ln|1+y|} = e^{Ln|x|+C}\]
bahrom7893
  • bahrom7893
Rewrite and simplify: \[e^{Ln|1+y|} = e^{Ln|x|}*e^C\] \[1+y = Kx\]
bahrom7893
  • bahrom7893
(e to some constant is another constant so I just let that other constant be K) Now apply initial conditions: f(-1)=1: 1+1 = -1K; 2 = -K; K = -2
bahrom7893
  • bahrom7893
So your answer is: 1+y = -2x, or: y = -2x - 1 <= Final answer
bahrom7893
  • bahrom7893
Im taking a differential equations course! IT IS FUN!! P.S.: Please click on become a fan if I helped, I really want to get to the next level!! Thanks =)
anonymous
  • anonymous
why do you multiply by e^c
bahrom7893
  • bahrom7893
there's a property: \[A^{B+C} = A^B * A^C\], so in our case: \[e^{Ln|x|+C} = e^{\ln|x|} * e^C\]
bahrom7893
  • bahrom7893
and e to some constant is another constant so I said let that constant be K. Any other questions?
anonymous
  • anonymous
no and thanks for your help

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