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Thread: Complex coefficient of a differential equation

  1. #1
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    Complex coefficient of a differential equation

    Hello everyone

    I have been given a question of which i am not sure of how to show:

    Observe the following differential equation:

    $\displaystyle a_{0}\frac{d^{n}y}{dt^{n}}+a_{1}\frac{d^{n-1}y}{dt^{n-1}}+\dots+a_{n-1}\frac{dy}{dt}+a_{n}y=0$

    where $\displaystyle a_{0},\dots,a_{n} \in \mathbb{C}$.

    Does the above differential equation always have a real solution, besides $\displaystyle y(t) = 0$?

    I my self think that it doesn't always have a real solution beside $\displaystyle y(t) = 0$, but i am not sure how to prove/show that i does not.

    Can anyone help?

    Kind regards
    Krisly
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  2. #2
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    Re: Complex coefficient of a differential equation

    Consider solutions to the characteristic equation when the $\displaystyle a_I$ are real, thinking in particular about complex conjugate pairs.

    Now compare with the case when he $\displaystyle a_I$ are complex.
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  3. #3
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    Re: Complex coefficient of a differential equation

    Thank you

    Although i am still not sure because, when it is for a differential equation with real coefficients, i get that the real solution is then given by:

    $\displaystyle y(t) = c_{1}e^{\alpha t} \cos{\beta t} + c_{2}e^{\alpha t} \sin{\beta t} $

    Considering one of the complex roots out of a pair of a complex conjugates.

    But when it comes to the situation where the coefficients are complex, then the roots does not necessarily come in those pairs, and that's where i get stuck

    Kind regards
    Krisly
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  4. #4
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    Re: Complex coefficient of a differential equation

    $\displaystyle c_1\mathrm e^{(\alpha +\mathrm i\beta)t}+c_2\mathrm e^{(\alpha -\mathrm i\beta)t} = \mathrm e^{\alpha t}\big((c_1+c_2)\cos(\beta t) + (c_1-c_2)\mathrm i \sin(\beta t)\big)$ and since $\displaystyle \mathrm I$ is a constant we get $\displaystyle c_1\mathrm e^{(\alpha +\mathrm i\beta)t}+c_2\mathrm e^{(\alpha -\mathrm i\beta)t} = \mathrm e^{\alpha t}\big(A\cos(\beta t) + B\sin(\beta t)\big)$. This highlights where the result you quoted comes from. When we are looking for real solutions, we will find that $\displaystyle c_1 $ and $\displaystyle c_2 $ are also complex conjugate pairs so that $\displaystyle A $ and $\displaystyle B $ are real.

    So, how much of the above can be guaranteed in the complex space? Clearly the values of $\displaystyle c_1 $ and $\displaystyle c_2 $ are arbitrary, so the question is: are we always able to find a pair of complex constants $\displaystyle c_1 $ and $\displaystyle c_2 $ that annihilate the imaginary part of the solution if we don't always have both $\displaystyle \mathrm e^{(\alpha +\mathrm i\beta)t}$ and $\displaystyle \mathrm e^{(\alpha -\mathrm i\beta)t}$ as solutions?
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  5. #5
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    Re: Complex coefficient of a differential equation

    As an alternate approach: consider the case $\displaystyle n=1$.
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