Answers to 2.4, 2.9, 2.14 and 2.19. I am not sure which differential equations are being referred to in 2.4 but assume they are (P.1) on page 83 together with e(p,u)=y.
2.4(a)We have de(p,u)/dp=x(p,e(p,u)) (a vector) and budget balancedness gives p.x(p,y)=y. Then p.de(p,u)/dp=p.x=y=e(p,u). Now use Euler's Theorem.
2.4(b)If e(p,u) is homogenous of degree one in p then Theorem A2.6 implies de(p,u)/dp is homogeneous of degree zero in p. Then de(tp,u)/dp=de(p,u)/dp so x(tp,e(tp,u))=x(p,e(p,u)) and hence x(tp,ty)=x(p,y) as e(tp,u)=te(p,u)=ty since e(p,u) is homogeneous of degree one in p. (Actually, you have to tweak Theorem A2.6 a little.)
2.9(a)Homogeneity of degree zero gives minus p2.dx2/dp2=p1.dx2/dp1+y.dx2/dy. Budget balancedness (BB) gives p1.x1+p2.x2=y so differentiating with respect to (wrt) p2 gives p1.dx1/dp2=-x2-p2.dx2/dp2=-x2+p1.dx2/dp1+y.dx2/dy(*). Now differentiate (BB) wrt y and multiply thru by x2 to get p1.x2.dx1/dy=x2-p2.x2.dx2/dy (**). Add (*) and (**), cancel p1>0 and use (BB) again to get dx1/dp2+x2.dx1/dy =dx2/dp1+x1.dx2/dy. This is the required condition.
2.9(b) Budget balancedness (BB) and WARP imply x(p,y) is homogeneous of degree zero in (p,y) (page 88). So by part (a), with two goods, BB and WARP imply that the Slutsky matrix associated with x(p,y) is symmetric. WARP and BB imply the Slutsky matrix is negative semidefinite (pages 88-90). So two goods, BB and WARP imply symmetry, negative semidefinite and BB. So by Theorem 2.6 x(p,y) is generated by utility maximizing behaviour. This means that R has no intransitive cycles.
2.14 A={a1,a2,...,an}. Use G1 to make a series of pairwise comparisons and construct the ascending series b1<=b2...<=bk<=...<=bn. Next use G2 to rule out the possibility of an infinitely ascending chain, that is to rule out bn<=b1.
2.19 An individual is risk averse if and only if (iff) u(pw1+(1-p)w2)>pu(w1)+(1-p)u(w2)iff u is strictly concave (pages 105 and 445).