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Math Help - Integration by Substitution

  1. #1
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    Integration by Substitution

    I would really appreciate it if someone could help me.

    Solve the differential equation by separation of variables.
    dy/dx = (y+5)(x+2)

    Thanks.
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  2. #2
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    Quote Originally Posted by turtle View Post
    Solve the differential equation by separation of variables.
    dy/dx = (y+5)(x+2)
    We have to solve on some open interval,
    y'=(y+5)(x+2).
    We want to divide by y+5.
    But we have to consider the case where y+5=0 in that case y=-5.
    By substitution it does solve the differencial equation.
    Second case consider y+5\not =0*
    And divide through,
    \frac{y'}{y+5}=x+2
    \int \frac{y'}{y+5} dx=\int x+2 dx
    \ln |y+5|=\frac{1}{2}x^2+2x+C_1
    |y+5|=\exp (1/2x^2+2x+C_1)=C\exp(1/2x^2+2x), C>0
    Thus,
    y+5=\pm C \exp(1/2 x^2+2x)=C\exp (1/2 x^2+2x), C\not =0
    y=-5+C\exp (1/2x^2+2x), C\not =0
    These are the necessary solutions, check them to show they all work.

    *)Note there is a case where y+5=0 for some point in the open interval and non-zero in some point. If that where the case then since the function is continous there is an open interval where it is non-zero and hence the solution we have above. But then there is no way that the function will connect will the zero point for that will lead to non-continuity and hence non-differenciability. But that cannot be the case because we assume y is differenciable on the open interval. Thus that cannot be the case.
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