2.14 only please

dt 1 + EU where u(0) = 1 and y is a positive constant. (a) For small e, find a two-term expansion of the solution (b) Solve the IVP and show that the solution satisfies In u + eu = y + . The is an example of an implicitly defined solution, which are very common when solving nonlinear differential equations. (c) Using the result from part (b), find a two-term expansion of u for small 2.12 As shown in Exercise 1.19, the nondimensional problem for the velocity of sphere dropping through the atmosphere satisfies du dt = -1 -u+eu? where u(0) = 0. (a) For small e, find a two-term expansion of the solution. , (b) Solve the IVP for u(t), and then derive a two-term expansion of it for smalle (c) Transform back into dimensional variables and obtain an expansion for vir). 2.13 The projectile problem that includes air resistance is dx + dt2 dr 1 (1 + 8x)? a where x(0) = 0 and 40) = 1. For small e, find a two-term expansion of the solution 2.14 Air resistance is known to depend nonlinearly on velocity, and the dependence is often assumed to be quadratic. Assuming gravity is constant, the equations of motion are 95 2 day dy dt dx dt 2 + (dy dt d12 c) + dex di2 dx -1-E- dt (dx dt + dy dt Histe is the vertical elevation of the object, and y is its horizontal location. The mial conditions are x(0) = y(0) = 0, and (o) = (0) = 1. The assumption is hat air resistance is weak, and so e is small and positive. For small , find the first terms in the expansions for x and y. Find the second terms in the expansions for x and y. 2.15 Consider the nonlinear boundary value problem G - y 0 y = 0, for 0