all parts please

2. (a) Let and be positive constants. Show that the heat equation tT=x22T has complex-valued solutions of the form F(x)eit provided F=iF. Hence find F if F(x)0 as x and F(0)=T1, where T1 is a positive constant. [You may assume that the roots of 2=i/ are =(1+i)/2.] (b) Now let T(x,t)=T0+Re(F(x)eit), where T0 is a real constant. Verify that T(x,t)=T0+T1exp(2x)cos(t2x), and explain why T(x,t) is a solution of the heat equation for which Tx(x,t)0 as x and T(0,t)=T0+T1cos(t). (c) A root cellar is used to store crops, ideally by keeping them as cool as possible in the summer, but as warm as possible in the winter. Consider a root cellar buried in soil of thermal diffusivity =106m2s1. Use the temperature profile in part (b) to predict (i) the shallowest ideal depth of the root cellar; (ii) the factor by which the amplitude of the temperature oscillations at ground level are reduced at the shallowest ideal depth