Question

Find a formula involving integrals for a particular solution of the differential equation $$ y^{\prime \prime \prime}-y^{\prime \prime}+y^{\prime}-y=g(t) $$

Short answer

Answer: The expression for the particular solution of the given differential equation is: $$ y(t) = \frac{1}{6}\int \int g(t) dt^3 + \frac{1}{2}C_1t^2 + C_2t + C_3 - \frac{1}{2}\int y(t) dt + \frac{1}{6}\int \int y(t) dt^3. $$

Step-by-step solution

Rewrite the equation

Rewrite the given differential equation as: $$ y^{\prime \prime \prime}(t)-y^{\prime \prime}(t)+y^{\prime}(t)-y(t)=g(t) $$

Integrate both sides

Integrate both sides of the equation to find y'(t): $$ \int [y^{\prime \prime \prime}(t)-y^{\prime \prime}(t)+y^{\prime}(t)-y(t)] dt = \int g(t) dt $$ This gives: $$ y^{\prime \prime}(t) - y^{\prime}(t) + y(t) - \int y(t) dt = \int g(t) dt + C_1 $$ where \(C_1\) is the constant of integration.

Integrate both sides again

Integrate both sides of the equation again to find y(t): $$ \int [y^{\prime \prime}(t) - y^{\prime}(t) + y(t) - \int y(t) dt] dt = \int [\int g(t) dt + C_1] dt $$ This gives: $$ y^{\prime}(t) - y(t) + \int y(t) dt - \frac{1}{2}\int \int y(t) dt^2 = \frac{1}{2}\int \int g(t) dt^2 + C_1t + C_2 $$ where \(C_2\) is another constant of integration.

Integrate both sides one last time

Integrate both sides of the equation a final time to find the particular solution for y(t): $$ \int [y^{\prime}(t) - y(t) + \int y(t) dt - \frac{1}{2}\int \int y(t) dt^2] dt = \int [\frac{1}{2}\int \int g(t) dt^2 + C_1t + C_2] dt $$ This gives: $$ y(t) + \frac{1}{2}\int y(t) dt - \frac{1}{6}\int \int y(t) dt^3 = \frac{1}{6}\int \int g(t) dt^3 + \frac{1}{2}C_1t^2 + C_2t + C_3 $$ where \(C_3\) is the last constant of integration. Now we have a formula involving integrals for a particular solution of the given differential equation: $$ y(t) = \frac{1}{6}\int \int g(t) dt^3 + \frac{1}{2}C_1t^2 + C_2t + C_3 - \frac{1}{2}\int y(t) dt + \frac{1}{6}\int \int y(t) dt^3 $$

Key concepts

Particular Solution

Finding a particular solution to a differential equation means identifying a specific solution that satisfies the equations given conditions. This is crucial, as differential equations often have a multitude of possible solutions, but a particular solution uses conditions such as initial values to narrow down the possibilities to a singular answer.
This process is akin to solving a puzzle where only certain pieces fit perfectly according to the problem's unique boundaries. In our case, we searched for a particular solution among many for the differential equation.

• Identifies a solution among many potential solutions.
• Uses conditions like initial values to uniquely determine the solution.

Once the particular solution is found, it yields a clearer understanding of the system modeled by the equation.

Integration

Integration is a fundamental operation in calculus and is pivotal in finding solutions to differential equations. It essentially reverses the operation of differentiation, turning rates of change back into original quantities. For our differential equation, integration was used multiple times. Each integration step allowed us to peel back layers of derivatives represented in the equation's terms of higher-order derivatives. This helped us arrive at a more basic form that we could use to construct the particular solution.

• Integrates functions to reverse differentiation.
• Allows for the construction of solutions based on equations that involve changing rates.
• Multiple integration steps can vastly simplify complex equations.

With each integration, we integrated from derivatives to get closer to a final form of the function, accumulating constants of integration along the way.

Constants of Integration

Whenever we perform an integration, we introduce an element known as the constant of integration. This arises because when we differentiate a constant, it disappears, making it impossible to know if an original function had one prior to differentiation without additional information.In this exercise, each integration step introduced a new constant, denoted as \(C_1, C_2,\) and \(C_3\). These constants are crucial as they adjust the solution to fit specific initial conditions or boundary values, tailoring the general integral solution into a particular one.

• Appear after each integration process.
• Essential for adjusting solutions to specific condition needs.
• Help define the exact form of the particular solution in conjunction with initial values.

Without addressing these constants, the solution could remain too broad, encapsulating more than just the targeted solution.

Higher-Order Derivatives

Higher-order derivatives are derivatives of derivatives, and they play a significant role in the behavior of differential equations, especially when these equations model complex systems.
In this exercise, the differential equation contains up to the third derivative of \(y\). This means the equation not only describes the function \(y\) but also its rate of change and the rate at which this change itself changes, all the way to its third order.

• Provide deeper insights into functions' behaviors and dynamics.
• Common in equations that describe systems with accelerating changes, like physical systems affected by forces.
• Influence the complexity of finding a particular solution.

Understanding higher-order derivatives is essential to unraveling more intricate models and exploring how changes at those higher levels affect the entire system.