Question

Consider the nonhomogeneous differential equation $$ y^{\prime \prime \prime}+a y^{\prime \prime}+b y^{\prime}+c y=g(t) \text {. } $$ In each exercise, the general solution of the differential equation is given, where \(c_{1}, c_{2}\), and \(c_{3}\) represent arbitrary constants. Use this information to determine the constants \(a, b, c\) and the function \(g(t)\). $$ y=c_{1}+c_{2} t+c_{3} e^{2 t}+4 \sin 2 t $$

Short answer

Question: Determine the coefficients a, b, and c, as well as the forcing function g(t) for the following nonhomogeneous linear third-order differential equation, given the general solution y = c1 + c2t + c3e^(2t) + 4sin(2t): y''' + ay'' + by' + cy = g(t) Answer: The coefficients for the given nonhomogeneous differential equation are a = 0, b = -1, and c = 1, and the forcing function g(t) = -24cos(2t) + 4sin(2t).

Step-by-step solution

Differentiate the General Solution

Differentiating the given general solution \(y\) with respect to time \(t\) to find \(y'\), \(y''\), and \(y'''\). $$ y = c_1 + c_2t + c_3e^{2t} + 4\sin(2t) $$$$ y' = c_2 + 2c_3e^{2t} + 8\cos(2t) $$$$ y'' = 4c_3e^{2t} - 16\sin(2t) $$$$ y''' = 8c_3e^{2t} - 32\cos(2t) $$

Substitute General Solution and Derivatives

Plug the general solution \(y\) and its derivatives \(y'\), \(y''\), and \(y'''\) back into the given differential equation, \(y'''+ay''+by'+cy=g(t)\): $$ (8c_3e^{2t} - 32\cos(2t)) + a(4c_3e^{2t} - 16\sin(2t)) + b(c_2 + 2c_3e^{2t} + 8\cos(2t)) + c(c_1 + c_2t + c_3e^{2t} + 4\sin(2t)) = g(t) $$

Solve the Equation for Coefficients

Comparing similar terms present in the equation, we can determine the coefficients \(a\), \(b\), and \(c\): $$ ae^{2t}=0 \quad\Rightarrow a=0 $$$$ 16b\cos (2t) + 32\cos (2t)=-32\cos (2t) \quad\Rightarrow b=-1 $$$$ 16c\sin(2t)-4\sin (2t) = 4\sin (2 t)\quad\Rightarrow c=1 $$ So, we find the coefficients to be \(a=0\), \(b=-1\), and \(c=1\).

Determine the Function \(g(t)\)

Now, we substitute the coefficients into the equation and find the function \(g(t)\): $$ g(t) = y''' + a\cdot y'' + b\cdot y' + c\cdot y = y''' - y' + y $$ Substituting our original expression for \(y, y'\), and \(y'''\) calculated before: $$ g(t) = (8c_3e^{2t} - 32\cos(2t)) - (c_2 + 2c_3e^{2t} + 8\cos(2t)) + (c_1 + c_2t + c_3e^{2t} + 4\sin(2t)) $$ The arbitrary constants \(c_1, c_2\) and \(c_3\) do not contribute to the forcing function \(g(t)\), therefore: $$ g(t) = -24\cos(2t) + 4\sin(2t) $$ #Conclusion# The coefficients for the given nonhomogeneous differential equation are \(a=0\), \(b=-1\), and \(c=1\), and the forcing function \(g(t) = -24\cos(2t) + 4\sin(2t)\).

Key concepts

General Solution

In the context of differential equations, the general solution plays a crucial role, particularly in nonhomogeneous forms. A differential equation is called nonhomogeneous when it includes a forcing function, which means it doesn't equate to zero. The general solution refers to the complete solution of such an equation, composed of two essential parts: the complementary solution (or homogeneous solution) and a particular solution.

• The complementary solution addresses the homogeneous part of the equation, which results from setting the non-homogeneous function to zero.
• The particular solution is specific to the non-homogeneous equation itself and depends on the function forcing the equation, denoted as \( g(t) \).

For example, for the differential equation \( y''' + ay'' + by' + cy = g(t) \), the general solution is provided as \( y = c_1 + c_2 t + c_3 e^{2t} + 4 \sin 2t \). This solution incorporates arbitrary constants \( c_1, c_2, \) and \( c_3 \), offering a family of functions that solve the equation for different initial conditions.

Arbitrary Constants

Arbitrary constants in the general solution, namely \( c_1, c_2, \) and \( c_3 \), are vital in tailoring solutions to meet initial conditions or specific circumstances. These constants represent the indefinite integral outcomes when a differential equation is resolved.

• They provide flexibility, allowing the solution to adjust based on early conditions given in a specific problem.
• Arbitrary constants reflect the infinite number of potential trajectories or solutions an equation might take in a solution space.

When working through the process of finding a solution to a differential equation, one determines these constants by applying initial conditions. These are the values of the function and its derivatives at a particular point. In scenarios where no specific initial conditions are provided, the general solution remains dependent on these constants.

Differentiating Functions

Differentiating functions is a key process in arriving at the general solution of a differential equation, especially within nonhomogeneous forms. Through differentiation, we obtain various orders of derivatives which are then substituted back into the original equation to verify solutions or derive parameters, such as constants and coefficients.

• Differentiate to find \( y', y'' \) and \( y''' \), which are essential when plugging into higher-order differential equations like \( y''' + ay'' + by' + cy = g(t) \).
• For example, the given general solution \( y = c_1 + c_2t + c_3e^{2t} + 4 \sin(2t) \) involves performing successive differentiation to find expressions for first, second, and third derivatives:
• These allow one to produce \( y' = c_2 + 2c_3e^{2t} + 8\cos(2t) \), \( y'' = 4c_3e^{2t} - 16\sin(2t) \), and \( y''' = 8c_3e^{2t} - 32\cos(2t) \).

By substituting back into the differential equation, you ensure consistency with the general solution, and it aids in finding unknown parameters.

Forcing Function

The forcing function, \( g(t) \), is what differentiates a nonhomogeneous differential equation from a homogeneous one. It represents external influences or inputs applied to the system described by the differential equation. Understanding and resolving for this component is essential in tackling real-world dynamic systems.

• In nonhomogeneous differential equations, \( g(t) \) dictates the behavior of solutions outside the response to initial conditions.
• It is critical in forming the particular solution component of the general solution, reflecting how the system responds to the external factors.

In our example, determining \( g(t) = -24\cos(2t) + 4\sin(2t) \) reveals the influences affecting the system. This derivation involves substituting known solutions back into the original differential equation format and solving for unmatched terms that contribute to \( g(t) \). Forcing functions are fundamental in applications where external stimuli drive system changes, informing everything from mechanical oscillations to electrical circuits.