Solution to differential equations using Taylor's series

Solution to differential equations using Taylor’s series

Understanding the Basics of Taylor’s Series for Differential Equations

• Taylor’s series are a powerful tool for approximating solutions to differential equations.
• A Taylor’s series is essentially a polynomial representation of a function, which is especially useful when the function itself is hard to deal with.
• For a function f(x), its Taylor’s series about the point x=a can be written as: f’(a) * (x-a) + f’‘(a) * (x-a)²/2! and so on.
• When using Taylor’s series, keep in mind that the approximate solutions will be less accurate the further they move from the point of expansion.
• Differential equations often present complex problems in mathematics. Sometimes their solutions cannot be expressed using elementary functions, and that’s where Taylor’s series come in.

Deriving a Taylor’s Series Solution for First Order Differential Equations

• Given a first order differential equation with an initial condition at x=a, the solution standard form is: y = f(a) + f’(a) * (x-a) + f’‘(a) * (x-a)²/2! and so on.
• To solve, first calculate as many derivatives of y as needed to match the order of the differential equation.
• Substitute the derivatives into the Taylor series expansion to derive a series representation of the solution.
• For example, if you have dy/dx + p(x)y = g(x), to find the Taylor series representation of the solution, you would first calculate derivatives of y, and then substitute them into the Taylor series formula.

Finding Taylor’s Series Solution for Higher Order Differential Equations

• Higher order differential equations are understandably more complex, but Taylor’s series can be just as effective in handling them.
• First, represent every term and derivative of y in the differential equation as an expression involving higher order derivatives at the point of expansion.
• Substitution into a Taylor series formula will then yield a series representation of the solution.

Important Aspects to Remember

• Be mindful of convergence. A Taylor Series can provide meaningful results provided it converges in the region of interest.
• An increase in the number of terms in the series will usually present a better approximation to the function.
• Taylor’s series can simplify the solution of differential equations and allow visualisation of the behaviour of functions.
• Mastery of the process - including the ability to find higher derivatives, keeping track of factors, and knowing how to manipulate power series - is essential for success in this method.