Why did Cauchy insist on proving that differential equations have solutions?

Augustin-Louis Cauchy insisted on proving that differential equations have solutions because it is not a priori obvious that every ordinary differential equation possesses a solution.1

His focus on existence proofs was a central component of his broader program to establish rigorous foundations for all of calculus. By rejecting the assumption that functions necessarily behave in ways that guarantee solutions, Cauchy sought to move mathematics away from reliance on unproven intuitions—such as those involving geometric or dynamic motion—and toward a formal, analytical framework based on continuity and limits. This shift was essential for transforming calculus into the modern, rigorous discipline of mathematical analysis.12