Differential equations


a. Find a power series for the solution of the following differential equations, subject to the given initial condition
b. Identify the function represented by the power series.


y′(t) − y = 0, y(0) = 2

Verified step by step guidance

1. Rewrite the differential equation in a form suitable for power series substitution: \(y'(t) - y(t) = 0\) with initial condition \(y(0) = 2\).
2. Assume a power series solution of the form \(y(t) = \sum_{n=0}^{\infty} a_n t^n\), where \(a_n\) are coefficients to be determined.
3. Differentiate the power series term-by-term to find \(y'(t) = \sum_{n=1}^{\infty} n a_n t^{n-1}\).
4. Substitute \(y(t)\) and \(y'(t)\) back into the differential equation to get \(\sum_{n=1}^{\infty} n a_n t^{n-1} - \sum_{n=0}^{\infty} a_n t^n = 0\).
5. Align powers of \(t\) by shifting indices as needed, then equate coefficients of like powers of \(t\) to form a recurrence relation for \(a_n\). Use the initial condition \(y(0) = a_0 = 2\) to find the coefficients.

Key Concepts

Power Series Solutions to Differential Equations

A power series solution expresses the solution of a differential equation as an infinite sum of powers of the independent variable. This method involves assuming a solution in the form of a series and determining the coefficients by substituting into the differential equation. It is especially useful when closed-form solutions are difficult to find.

Initial Conditions and Their Role

Initial conditions specify the value of the solution and possibly its derivatives at a particular point, allowing us to find unique coefficients in the power series. For example, y(0) = 2 sets the constant term in the series, ensuring the solution matches the given starting value.

Identifying Functions from Power Series

Once a power series solution is found, it can often be recognized as a known function by comparing it to standard series expansions (e.g., exponential, sine, cosine). This identification helps express the solution in a closed form, making it easier to interpret and use.