During the manufacture of a steel component it is often necessary to quench them in a large bath of liquid in order to cool them down. This reduces the temperature of the component to the temperature of the liquid. If T is the temperature of the component in excess of the liquid temperature, the rate of change of the component proportion to the temperature of the component. Take the proportional constant as K. K depends upon the volume and surface area of the component, its specific heat capacity, and the heat transfer coefficient between the component and the liquid.

a) Formulate a differential equation for the above engineering solution

b) Determine a solution for the formulated formula using numerical methods and power series, the initial condition that t+0 the temperature excess is 200.