401 Initial Conditions
A value of a function at a fixed point of its domain that is used to determine a unique solution of an equation.
The initial condition. An initial condition is the fixed value that the unknown solution must equal at a given starting value of the independent variable. This principle is used to pin a solution curve to one starting point.
The initial condition is
\[ y(t_{0}) = y_{0} \]
where
- \(y\) is the unknown function.
- \(t_{0}\) is a fixed value of the independent variable.
- \(y_{0}\) is the prescribed value of \(y\) at \(t_{0}\).
The initial-value problem. An initial-value problem is a differential equation together with an initial condition. This principle is used to select one physical solution from the family of general solutions obtained by integrating the equation.
Picard’s existence and uniqueness theorem. If the slope function and its partial derivative are continuous near the starting point, a unique solution exists nearby. A Lipschitz condition is a smoothness bound that keeps the solution from splitting into many paths. This principle is used to guarantee uniqueness before one tries to find the solution.
Note: Also called an initial-value problem when the differential equation and the initial condition are taken together. Picard’s existence and uniqueness theorem is also called Picard’s theorem.
401.1 References
- Stewart, J. Calculus: Early Transcendentals. — initial condition \(y(t_{0})=y_{0}\); initial-value problem.
- Logan, J. D. A First Course in Differential Equations. — the initial-value problem as equation plus starting value.
- Simmons, G. F. Differential Equations with Applications and Historical Notes. — Picard’s existence and uniqueness theorem.
- Kreyszig, E. Advanced Engineering Mathematics, 10th ed. Wiley, 2011. — Picard existence and uniqueness.