mfb said:As long as all those different y'(0) lead to valid solutions, that works.
The solution space in ODEs refers to the set of all possible solutions to a given ordinary differential equation. It is often represented as a graph or a geometric shape in which different initial conditions can lead to different solutions.
The solution space can be represented in various ways, such as graphs, geometric shapes, or vector fields. These representations help visualize the behavior of the solutions and provide insights into the dynamics of the system described by the ODE.
The size and shape of the solution space can be affected by several factors, including the coefficients and initial conditions of the ODE, the order and linearity of the equation, and the presence of boundary conditions. Changes in these factors can lead to different solution spaces.
The solution space can be utilized to determine the existence and uniqueness of solutions, as well as to find specific solutions that satisfy given initial conditions. It can also help identify patterns and behaviors in the solutions, which can aid in understanding the system described by the ODE.
Yes, there are some limitations to the solution space in ODEs. For example, the solution space may not be defined for all values of the independent variable, or it may contain regions where the solution is not smooth or continuous. Additionally, the solution space may not accurately represent the behavior of the solutions for highly nonlinear or complex ODEs.