Two students investigate the first–order differential equation dxdy=f(x), where f is continuous on an interval I.
The students obtain two antiderivatives of f that differ by a non-zero constant and wonder whether both expressions represent the same family of solutions.
Answer the following, clearly labeling each response.
(a) Describe, in your own words, why any two antiderivatives of the same continuous function must differ by a constant and therefore represent the same one-parameter family of curves.
(b) Explain how supplying a single initial condition y(a)=y0 determines a unique curve from that family, and identify the geometric property of the graph that is fixed by choosing the constant accordingly.
(c) Write a formula for the particular solution that passes through (a,y0) using a definite integral with variable upper limit, and interpret the meaning of each symbol in that formula.
(d) Discuss how the domain of a particular solution can be more restrictive than the domain of the general family. Illustrate your discussion with the example f(x)=x1, paying particular attention to the role of ∣x∣ in the antiderivative.