Question

The general solution of the nonhomogeneous differential equation \(y^{\prime \prime}+\alpha y^{\prime}+\beta y=g(t)\) is given, where \(c_{1}\) and \(c_{2}\) are arbitrary constants. Determine the constants \(\alpha\) and \(\beta\) and the function \(g(t)\). $$y(t)=c_{1} e^{t} \cos t+c_{2} e^{t} \sin t+e^{t}+\sin t$$

Short answer

Answer: \(\alpha = 1\), \(\beta = 1\), and \(g(t) = e^t - \sin t + \cos t\).

Step-by-step solution

Identifying the homogeneous and particular solutions

We are given the general solution: $$y(t)=c_{1}e^{t} \cos t+c_{2}e^{t} \sin t+e^{t}+\sin t$$ The homogeneous part of the solution is: $$y_h(t)=c_{1}e^{t} \cos t+c_{2}e^{t} \sin t$$ And the particular solution is: $$y_p(t)=e^{t}+\sin t$$

Find the first and second derivatives of the general solution

To determine \(\alpha\), \(\beta\) and \(g(t)\), we need to find the first and second derivatives of the general solution. First derivative, \(y'(t)\): $$y'(t)=c_{1}e^{t}(\cos t-t\sin t)+c_{2}e^{t}(\sin t+t\cos t)+e^{t}+\cos t$$ Second derivative, \(y''(t)\): $$y''(t)= c_1 e^{t} (\sin t \cdot (-t-1)+\cos t \cdot(-t)) + c_2 e^{t}(\cos t \cdot (t+1)+\sin t \cdot t) + e^{t} - \sin t$$

Plug the derivatives into the nonhomogeneous differential equation

Now, substituting the given general solution and its derivatives into the nonhomogeneous differential equation: $$y''(t) + \alpha y'(t) + \beta y(t) = g(t)$$ Replace the expressions for \(y(t)\), \(y'(t)\), and \(y''(t)\): $$c_1 e^{t} (\sin t \cdot (-t-1) + \cos t \cdot (-t)) + c_2 e^{t} (\cos t \cdot (t+1) + \sin t \cdot t) - \sin t + e^{t}\\ + \alpha[c_{1}e^{t}(\cos t-t\sin t)+c_{2}e^{t}(\sin t+t\cos t)+e^{t}+\cos t]\\ + \beta[c_{1}e^{t} \cos t+c_{2}e^{t} \sin t+e^{t}+\sin t]=g(t)$$

Match the coefficients to determine \(\alpha\), \(\beta\) and \(g(t)\)

Notice that: - The coefficient of \(c_1 e^{t} \cos t\) in the above expression should be \(0\) because it belongs to the homogeneous solution. This gives us the equation: $$-\alpha t + \beta =0$$ - The coefficient of \(c_2 e^{t} \sin t\) in the above expression should also be \(0\) because it belongs to the homogeneous solution. This gives us the equation: $$\alpha t + \beta =0$$ Solving these two equations, we get: $$\alpha = 1$$ $$\beta = 1$$ Now, the function \(g(t)\) will be the remaining part of the above expression after removing the terms containing \(c_1\) and \(c_2\). So, we get: $$g(t) = e^t - \sin t + \cos t$$ We have now determined the constants \(\alpha\), \(\beta\) and the function \(g(t)\). In conclusion: $$\alpha = 1$$ $$\beta = 1$$ $$g(t) = e^t - \sin t + \cos t$$

Key concepts

General Solution

The general solution of a nonhomogeneous differential equation combines both the homogeneous and particular solutions. It represents the broadest possible form of the solution which satisfies the differential equation. For example, in the equation given, the general solution is expressed as:

• \( y(t) = c_1 e^{t}\cos t + c_2 e^{t} \sin t + e^t + \sin t \)

Here, \(c_1\) and \(c_2\) are arbitrary constants that allow the solution to incorporate any initial conditions. This general form is crucial because it gives us the flexibility to match the unique requirements of a given problem by adjusting these constants. Adding the particular solution to the homogeneous solution gives us the general solution, providing a comprehensive way to tackle the complexity of nonhomogeneous systems.

Homogeneous Solution

The homogeneous solution is the part of the solution that solves the associated homogeneous differential equation, which is obtained by setting the nonhomogeneous term to zero. How these solutions form is closely tied to the nature of the differential equation and reflects the behavior of the system devoid of external forces or inputs. For the equation provided, the homogeneous solution is:

• \( y_h(t) = c_1 e^{t}\cos t + c_2 e^{t}\sin t \)

This represents the natural response of the system without external influence (i.e., the right-hand side \(g(t)\) of the original equation is zero). It captures the manner in which the system might oscillate or exponentially grow or decay intrinsically. The form is often determined using techniques like characteristic equations or matrices for higher-order systems. Homogeneous solutions are foundational to understanding the dynamic nature of linear differential equations.

Particular Solution

The particular solution is what gives us the ability to solve the nonhomogeneous differential equation. It is the component of the solution that specifically addresses the nonhomogeneous part or external input represented by the function \(g(t)\) in the differential equation. For the given problem:

• \( y_p(t) = e^{t} + \sin t \)

This particular solution directly corresponds to the external inputs or forces acting on the system. Techniques to find a particular solution vary, including the method of undetermined coefficients or variation of parameters. These methods help in crafting a solution that fits exactly with \(g(t)\), ensuring that when added to the homogeneous solution, we achieve the full general solution that satisfies the entire differential equation system. It's a crucial step in bridging the natural system behavior with real-world applications or perturbations.

Differential Equation System

Differential equation systems consist of one or multiple differential equations that are interconnected. These systems are used to model complex dynamics where multiple factors interact, often seen in fields such as physics, engineering, and economics. The problem at hand involves:

• A second-order differential equation of the form \(y'' + \alpha y' + \beta y = g(t)\).

Solving such systems involves finding solutions that satisfy all involved equations, typically through a combination of general, homogeneous, and particular solutions. These solutions are vital for predicting system behavior under different conditions. In our case, the system parameters \(\alpha\) and \(\beta\) were found by equating coefficients derived from inserting the solution back into the differential equation, allowing for exploration of system characteristics, including stability and response to varying inputs. Mastery of these equations unlocks understanding of how a system operates in entirety.