[i][b]Introduction:[br][/b][/i][br]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[br][br][math]\frac{dy}{dx}=f\left(x,y\right);y\left(0\right)=y_0[/math][br][br]if [math]f(x,y)[/math] and [math]\frac{d}{dy}f\left(x,y\right)[/math]are continuous at [math](0,y_0)[/math][b],[/b] then there exists a unique solution [math]\phi\left(x\right)[/math] such that [math]\phi\left(0\right)=y_0[/math][b].[br][br][/b]The proof of this statement hinges on the so-called[b] Picard's Method of Successive Approximations.[/b] 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 [b]Picard's Sequence of Approximate Solutions, [/b]and it can be shown that it converges to exactly one function, [math]y=\phi\left(x\right)[/math], of the independent variable. [br][br]In addition to its theoretical import, Picard's Method is also an alternative to numerical methods such as Euler's Method or RK4. [br][br][br][i][br][br][br][b]The Method:[/b][/i][br][br]Given a first order differential equation with initial value[br][br][math]\frac{dy}{dx}=f\left(x,y\right);y\left(0\right)=y_0[/math][br][br]Picard's Sequence of Successive Approximate Solutions is generated by[br][br][math]\phi_{n+1}\left(x\right)=\int_0^xf\left(s,\phi_n\left(s\right)\right)ds[/math][br][br]where [math]\phi_0\left(x\right)=y_0[/math]. A proof that this sequence converges to exactly to the solution function, [math]y=\phi\left(x\right)[/math] can be found in any standard text on differential equations. [br][br][br][br][br][br][i][b]About the Applet:[/b][/i][br][br]The applet below illustrates Picard's Sequence of Successive Approximate Solutions to the differential equation with initial value[br][br][math]\frac{dy}{dx}=2x\left(1+y\right);y\left(0\right)=0[/math][br][br]The exact specific solution can be found via the method of separation, and is pictured in purple. [br][br]The elements of Picard's Sequence are shown in green. [br][br]Slide the variable to see successively more accurate elements of Picard's Sequence converge to the exact specific solution.