# Linear ODE with Variable coefficients

• Aug 3rd 2013, 11:20 PM
spolley
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.
• Aug 4th 2013, 12:17 AM
Prove It
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...