Recent advancements in large language models have significantly expedited the process of generating front-end code. This allows users to rapidly prototype user interfaces and ideate through code, a process known as exploratory programming. However, existing LLM code generation tools focus more on technical implementation details rather than finding the right design given a particular problem. We present DynEx, an LLM-based method for design exploration in accelerated exploratory programming. DynEx introduces a technique to explore the design space through a structured Design Matrix before creating the prototype with a modular, stepwise approach to LLM code generation. Code is generated sequentially, and users can test and approve each step before moving onto the next. A user study of 10 experts found that DynEx increased design exploration and enabled the creation of more complex and varied prototypes compared to a Claude Artifact baseline. We conclude with a discussion of the implications of design exploration for exploratory programming.