> [!NOTE] Theorem > The solution to the following [[Initial Value Problem for Scalar Ordinary Differential Equation|initial value problem]]: the [[First Order Linear Ordinary Differential Equation|first order linear ODE]] $\frac{d}{dt}x(t)+p(t)x(t)=q(t)$subject to $x(t_{0})=x_{0}$; is given by $x(t)e^{P(t)}= x_{0} + \int_{t_{0}}^{t} q(\tilde{t})e^{P(\tilde{t})} \, d\tilde{t} $where $P(t)=\int p(t) \, dt.$ **Proof**: .... # Applications **Newton’s law of cooling**: [[Newton's law of cooling]] provides a mathematical model of the temperature $T(t)$ of an object in surroundings of temperature $A(t)$: $\frac{dT}{dt}=-k(T-A(t))$ where $k>0$ measures the rate of heat absorbed (or emitted) by the object. A forensic method [[Estimating the time of death|estimating the time of death of a body]] is based on Newton’s law of cooling: the idea is to take the temperature of the body at two different times, in order to give an estimate of the constant $k$ to be used in the equation, and then to extrapolate back to find the time when $T$ is the temperature of a living body, $37^{\circ}C$.