# Picard's Method of Successive Approximations

*After studying the various methods for solving and numerically estimating solutions to first order differential equations with initial values, you might wonder if there is any theory that informs the existence and uniqueness of the solutions you have found. The answer is a resounding "yes!" For a differential equation*

**Introduction:****,**then there exists a unique solution

**.**The proof of this statement hinges on the so-called

**Picard's Method of Successive Approximations.**Picard's Method generates a sequence of increasingly accurate algebraic approximations of the specific exact solution of the first order differential equation with initial value. The sequence is called

**Picard's Sequence of Approximate Solutions,**and it can be shown that it converges to exactly one function,

*Given a first order differential equation with initial value*

**The Method:***The applet below illustrates Picard's Sequence of Successive Approximate Solutions to the differential equation with initial value*

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