Question

a. Find a power series for the solution of the following differential equations. b. Identify the function represented by the power series. $$y^{\prime}(t)-3 y(t)=10, y(0)=2$$

Short answer

Based on the step by step solution, create a short answer question for the exercise: Question: Given the first-order linear differential equation $$y^{\prime}(t) = 3y(t) + 10$$ with the initial condition $$y(0) = 2$$, find the power series representation of the solution, and then identify the function that it represents. Answer: The power series representation of the solution is $$y(t) = 2 + 16t + 24t^2 + 24t^3$$, and the function it represents is $$y(t) = 2 + 16t + 24t^2 + 24t^3$$.

Step-by-step solution

Write the differential equation in the form of a recursion relation

We can rewrite the given differential equation as: $$y^{\prime}(t) = 3y(t) + 10$$ Now, let $$y(t) = \sum_{n=0}^{\infty}a_nt^n$$ be the power series representation of the solution. Then, we have $$y^{\prime}(t) = \sum_{n=1}^{\infty}na_nt^{n-1}$$ Equating both sides of the equation for the coefficients of like powers gives us a recursion relation: $$na_n = 3a_{n-1} + 10 \delta_{n,1}$$ where \(\delta_{n,1}\) is the Kronecker delta, which is 1 if n=1 and 0 otherwise. We also have the initial condition given as $$y(0) = \sum_{n=0}^{\infty}a_0 = 2$$

Solve the recursion relation to find the coefficients

Using the equation found in Step 1, we have: $$a_0 = 2$$ $$a_1 = 3a_0 + 10 = 16$$ $$a_2 = \frac{3}{2}a_1 = 24$$ $$a_3 = \frac{3}{3}a_2 = 24$$ Continuing with this pattern, we see that: $$a_n = 24\delta_{n,2} + 24\delta_{n,3}$$ for $$n \ge 2$$

Write the power series solution using the coefficients

We found the coefficients $$a_n$$ in the previous step, so we can now write the power series solution: $$y(t) = a_0 + a_1t + \sum_{n=2}^{\infty} a_nt^n = 2 + 16t + 24t^2 + 24t^3$$

Identify the function represented by the power series

The function represented by the power series can be determined by inspection or by comparing it to a known power series. In this case, the function can be identified through inspection as a polynomial. Hence, the function represented by the power series is: $$y(t) = 2 + 16t + 24t^2 + 24t^3$$

Key concepts

Differential Equations

A differential equation is a mathematical equation that relates a function with its derivatives. In simple terms, it shows how a certain quantity changes over time, often in response to another quantity. Differential equations are fundamental in describing various physical phenomena. They are used extensively in engineering, physics, economics, and other scientific fields.

In this specific exercise, we deal with a first-order linear differential equation given as: \[ y^{\prime}(t) - 3y(t) = 10 \] This equation tells us how the function \( y(t) \) and its rate of change \( y^{\prime}(t) \) are related. The additional 10 represents an external source that impacts the rate of change.

Understanding this is key because the solution we're looking for is not just any function, but one that satisfies this specific relationship.

Recursion Relation

A recursion relation is an important concept in mathematics. It is an equation that defines each term of a sequence based on the preceding terms. In the context of power series and differential equations, recursion relations help us find the coefficients necessary to build a solution.

For the differential equation in this problem, rewriting it as a power series leads us to this recursion relation: \[ na_n = 3a_{n-1} + 10 \delta_{n,1} \] This formula helps us determine each coefficient \( a_n \) needed for the power series representation of the solution. The Kronecker delta \( \delta_{n,1} \) is used here; it's a function that is 1 when \( n = 1 \) and 0 otherwise, ensuring correct application of the initial condition and the constant term.

Solving this recursion relation step by step gives us the coefficients such as \( a_0 = 2, a_1 = 16, \) and so on, ultimately leading us to construct the solution.

Power Series Representation

The power series representation is a way to express a function as an infinite sum of terms. Each term consists of a constant coefficient multiplied by a power of the variable. This approach is incredibly useful in situations where exact solutions are difficult or impossible to find explicitly. It approximates solutions and helps unravel complex functions into simpler components.

In our example, the function \( y(t) \) is represented as: \[ y(t) = \sum_{n=0}^{\infty}a_nt^n \] This means the solution is expressed as an infinite series of the powers of \( t \), where each coefficient \( a_n \) is determined based on the recursion relation.

The power series provides a flexible framework to approximate and analyze the behavior of differential equations in a more manageable format.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Such functions are simple, continuous, and easy to work with. They often perfectly model real-world phenomena over limited regions. Understanding polynomials is essential especially when identifying functions from power series solutions.

In the conclusion of our power series solution, the result was a polynomial: \[ y(t) = 2 + 16t + 24t^2 + 24t^3 \] This function is a cubic polynomial, as its highest term is to the power of three. Polynomials like this one clearly define their behavior and are neatly expressed, unlike infinite series. Therefore, converting a power series representation into such a polynomial form ends the problem-solving process neatly, providing a clear understanding of the function's behavior.

The advantage of polynomials lies in their simplicity and straightforward application, often making them the preferred form when possible.