The study of differential equations and the study of algebraic geometry are two disciplines within mathematics that seem to be mostly disjoint from each other. Looking deeper, however, one finds that connections do exist. This thesis gives in four chapters four examples of interesting mathematical insights that can be gained from combining the concepts and techniques from both of these fields. The first project shows how the behaviour of solutions of certain differential equations can be better understood by considering algebraic curves with a differential form. The second project proves the existence of certain higher differential operators in algebraic settings where these were not known to occur before. The third project shows that the existence or non-existence of power series solutions of partial differential equations can be interpreted from the perspective of tropical geometry. And the last project relates the old theorem of Siegel about integral points on elliptic curves to the monodromy of linear differential equations on this elliptic curve.