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Math Help - FInding particular solutions of differential equations

  1. #1
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    FInding particular solutions of differential equations

    I got stuck on these on my homework, and was hoping someone could help be out.

    1. ysqrt(1-x^2)y' - xsqrt(1-y^2) = 0
    Initial condition is y(0) = 1

    2. dr/ds = e^(r+s)
    Initial condition is r(1) = 0

    Thanks!
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  2. #2
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    Quote Originally Posted by alreadyinuse View Post
    I got stuck on these on my homework, and was hoping someone could help be out.

    1. ysqrt(1-x^2)y' - xsqrt(1-y^2) = 0 Mr F says: y \, \sqrt{1 - x^2} \, \frac{dy}{dx} = x \sqrt{1 - y^2} \Rightarrow \frac{y}{\sqrt{1 - y^2}} \, dy = \frac{x}{\sqrt{1 - x^2}} \, dx ......

    Initial condition is y(0) = 1

    2. dr/ds = e^(r+s) Mr F says: Writing e^{r + s} = e^r e^s might make the DE more obviously seperable to you .....

    Initial condition is r(1) = 0

    Thanks!
    ..
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  3. #3
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    Quote Originally Posted by alreadyinuse View Post
    I got stuck on these on my homework, and was hoping someone could help be out.

    1. ysqrt(1-x^2)y' - xsqrt(1-y^2) = 0
    Initial condition is y(0) = 1

    2. dr/ds = e^(r+s)
    Initial condition is r(1) = 0

    Thanks!

    The first is seperable...

    \frac{y}{\sqrt{1-y^2}}dy=\frac{x}{\sqrt{1-x^2}}dx

    Just integrate both sides

    The second is also seperable.

    note that

    e^{r+s}=e^re^s

    This gives

    e^{-r}dr=e^sds

    take the integral of both sides
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