Note: please provide all examples in each category of the venn diagram(7 in total).

a. Categorize each of the following PDE's according to order, homogeneity, autonomy, and linearity. (i) Ut + uur = xtu (ii) Utt Uxt = U-3 (iii) 3xUz + Utc = 0 b. In each region of the Venn diagram below, provide an example of a PDE that has all the properties which intersect in that region. Importantly, if a region excludes a property, you must ensure that your example does not have this property. (For example, in the region given strictly by the intersection of homogeneity and au- tonomy, your example PDE must be homogeneous and autonomous but not linear.) Homogeneous Autonomous Linear a. Categorize each of the following PDE's according to order, homogeneity, autonomy, and linearity. (i) Ut + uur = xtu (ii) Utt Uxt = U-3 (iii) 3xUz + Utc = 0 b. In each region of the Venn diagram below, provide an example of a PDE that has all the properties which intersect in that region. Importantly, if a region excludes a property, you must ensure that your example does not have this property. (For example, in the region given strictly by the intersection of homogeneity and au- tonomy, your example PDE must be homogeneous and autonomous but not linear.) Homogeneous Autonomous Linear