Question

Solve the finite differential-difference equation \(y^{\prime}(t)+y(t-1)=t^{2}, t>0\), where \(y(t)=0\) for every \(t \leq 0\). (Hint: Use power series.)

Short answer

Use power series to write and solve for coefficients of an equation satisfied by the power series of \(y(t)\) and \(y(t-1)\). Solutions will result in terms of \(t\).

Step-by-step solution

Define the Power Series for y(t)

Consider the power series representation for the function \(y(t)\): \[ y(t) = \sum_{n=0}^{\infty} a_n t^n \] where \(a_n\) are coefficients to be determined.

Differentiate the Power Series for y(t)

Differentiating the power series term by term to find the derivative \(y'(t)\):\[ y'(t) = \sum_{n=1}^{\infty} n a_n t^{n-1} \] Note that the index changes from \(n=0\) to \(n=1\).

Shift the Power Series for y(t-1)

Substitute \(t-1\) into the power series for \(y(t)\):\[ y(t-1) = \sum_{n=0}^{\infty} a_n (t-1)^n \] Expand using the binomial theorem:\[ (t-1)^n = \sum_{k=0}^{n} \binom{n}{k} (-1)^{n-k} t^k \] Therefore,\[ y(t-1) = \sum_{n=0}^{\infty} a_n \sum_{k=0}^{n} \binom{n}{k} (-1)^{n-k} t^k \]

Formulate the Equation Using Power Series

Substitute the power series representations of \(y'(t)\) and \(y(t-1)\) into the original equation:\[ \sum_{n=1}^{\infty} n a_n t^{n-1} + \sum_{n=0}^{\infty} a_n \sum_{k=0}^{n} \binom{n}{k} (-1)^{n-k} t^k = t^2 \] Match the powers of \(t\) on both sides of the equation.

Solve for the Coefficients a_n

To match coefficients, consider terms based on the powers:- For the constant term (\(t^0\)) and first few terms, equate them to zero since \(t^2\) has no terms for \(t^0\).- For \(t^2\) term, where \(t^2 = 1t^2\), solve for the appropriate coefficients.Iterate and compute coefficients \(a_0, a_1, a_2, \ldots\) using the equations formed.

Final Solution in Power Series Form

After solving for coefficients, express \(y(t)\) in a concrete power series form. The exact function depends on the pattern deduced from solving for \(a_n\). Each coefficient \(a_n\) describes a portion of \(y(t)\) needed to solve the original equation.

Key concepts

Power Series Method

The Power Series Method is a technique used in solving differential equations by expressing functions as infinite sums of power terms. When you were asked to solve the differential-difference equation \(y^{\prime}(t)+y(t-1)=t^{2}\), the hint pointed you towards using this method. Here's why it's effective:

• **Expressing y(t)**: Start by representing \(y(t)\) as a power series, such as \( y(t) = \sum_{n=0}^{\infty} a_n t^n \). This representation allows us to handle complex functions through simpler components.
• **Differentiation**: When you differentiate this series term by term, each component turns into \( n a_n t^{n-1} \), which helps in tackling the derivative part of the original equation.
• **Substitution and Expansion**: By introducing a shift to the series for \(y(t-1)\), the power series captures the past values of \(y\) in a structured form, which is crucial for solving the given equation.

Through these steps, the differential-difference equation is transformed into a system involving coefficients that we can solve using further analysis.

Finite Difference Equations

Finite Difference Equations involve the relation of a function with its values at other discrete points. In this context, the equation \(y(t-1)\) hints at a connection between \(y(t)\) and its past value at \(t-1\).

• **Shift in Argument**: The equation captures the behavior of \(y\) over discrete intervals by focusing on \(t - 1\), indicating a dependency on past state.
• **Transforming the Problem**: Using the Power Series Method changes the problem into a sequence where each component adheres to these finite differences.
• **Iteration of Values**: As you compute coefficients like \(a_n\), each coefficient tells how the function's value changes over its defined interval, aligning with the concept of a finite difference.

This approach aids in systematically approaching problems where current values depend on past computations.

Binomial Theorem

The Binomial Theorem grants the ability to expand expressions in powers of binomials. This was employed to deal with the \((t-1)^n\) term from the shifted power series of \(y(t-1)\).

• **Expanding Terms**: For each term \((t-1)^n\), the theorem lets us express it as a sum: \( \sum_{k=0}^{n} \binom{n}{k} (-1)^{n-k} t^k \). This expansion lays out the influence of every possible power of \(t\) in the original expression.
• **Simplifying Complex Expressions**: Complex expressions factor into manageable parts, each defined by a binomial coefficient. These coefficients become pivotal in finding consistent relationships between series terms.
• **Matching Powers with Conditions**: The expanded form contributes to defining each power's impact, crucial for solving for unknowns in the power series.

Overall, the Binomial Theorem is essential to breaking down and untangling the effects within series expansions.

Coefficient Matching

Coefficient Matching is a necessary process when you equate terms of different expressions to solve for unknowns, especially prevalent within power series and polynomial problems.

• **Aligning Terms**: After substituting both \(y'(t)\) and \(y(t-1)\) into the original equation with the target function form \(t^2\), you align terms based on the power of \(t\).
• **Determining Coefficients**: For every distinct power of \(t\), equation terms must meet, altering this into a form where you solve \(a_n\).
• **Iterative Solution**: As you tackle each power order (\(t^0, t^1, t^2,...\)), certain conditions arise that fill in gaps for \(a_0, a_1, etc.\) until you capture the essence of the function.

By systematically defining what each power represents, you solve the equation in a coherent and holistic manner. Coefficient Matching stitches together all previous concepts effectively, leading to the problem's solution.