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 = k_{1}exp(k_{2}t) - k_{3}(T-T_{0});

where k1, k2, k3 are positive constants. Determine T(t), assuming that T(1) = T_{0}.

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.

Re: Linear ODE with Variable coefficients

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

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