In solving differential equations, the computations can almost always be simplified by theuse of dimensionless variables.(a) Show that if the dimensionless variable = x/L is introduced, the heat conductionequation becomes2u2 = L22ut, 0 << 1, t > 0.(b) Since L2/2 has the units of time, it is convenient to use this quantity to define a dimensionlesstime variable = (2/L2)t. Then show that the heat conduction equationreduces to2u2 = u , 0 << 1, > 0.

L21 - 6 ex. Aditbedofitbyugi through an inverted cone-shaped ﬁlter. The height of the cone is 20 inches and the diameter across the top is 16 inches. If the liquid is ﬂowing out at 2 cubic in/min, how fast is the depth of the liquid changing when it is 12 inches deep