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Math Help - Linear ODE with Variable coefficients

  1. #1
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    Question Linear ODE with Variable coefficients

    A global warming model leads to the following equation for the average temperature of the
    Earth's surface:


    dT/dt = k1exp(k2t) - k3(T-T0);


    where k1, k2, k3 are positive constants. Determine T(t), assuming that T(1) = T0.

    Any assistance would be appreciated (:
    My lecturer did not explain how to solve these equations very well and I have tried searching all over the internet for help on solving these equations but I have been unsuccessful and/or lead to more confusion.
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  2. #2
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    Re: Linear ODE with Variable coefficients

    \displaystyle \begin{align*} \frac{dT}{dt}&= k_1\,e^{k_2\,t} - k_3 \left( T - T_0 \right) \\ \frac{dT}{dt} &= k_1 \, e^{ k_2\,t} - k_3\,T - k_3\,T_0 \\ \frac{dT}{dt} + k_3\,T &= k_1 \, e^{k_2\,t} - k_3\,T_0 \end{align*}

    This is now first-order linear, so you can use the Integrating Factor method to solve...
    Thanks from HallsofIvy
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