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Math Help - [SOLVED] Differential Equation

  1. #1
    Member ronaldo_07's Avatar
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    [SOLVED] Differential Equation

    Determine the general solution to the differential equation
    y′ = \frac{y}{x^2+4x+5}

    Fix the constant of integration according to the initial condition y(−1) = 1
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  2. #2
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    Quote Originally Posted by ronaldo_07 View Post
    Determine the general solution to the differential equation
    y′ = \frac{y}{x^2+4x+5}

    Fix the constant of integration according to the initial condition y(−1) = 1
     \frac{dy}{dx} = \frac{y}{x^2+4x+5}

     \frac{dy}{y} = \frac{dx}{x^2+4x+5}

     \frac{dy}{y} = \frac{dx}{(x+2)^2+1}

    Remember that  \int \frac{dx}{x^2+a^2} = \frac{1}{a}\arctan(\frac{x}{a}) + C , and  \int \frac{dx}{x} = \ln{|x|} + C
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  3. #3
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    Quote Originally Posted by ronaldo_07 View Post
    Determine the general solution to the differential equation
    y′ = \frac{y}{x^2+4x+5}

    Fix the constant of integration according to the initial condition y(−1) = 1
    \frac{dy}{dx} = \frac{y}{x^2+4x+5}

    \frac{dy}{y} = \frac{dx}{(x+2)^2 + 1}

    \ln|y| = \arctan(x+2) + C

    \ln(1) = \arctan(-1+2) + C

    0 = \arctan(1) + C

    C = -\frac{\pi}{4}

    finish up the solution
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  4. #4
    Member ronaldo_07's Avatar
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    to get y= shal I just take log on both sides so y= e^{arctan}......
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  5. #5
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    \ln|y| = \arctan(x+2) - \frac{\pi}{4}

    change to an exponential equation ...

    y = e^{\arctan(x+2) - \frac{\pi}{4}}
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