Question

Let \(u_{1}(x, t)\) be a solution of the linear homogeneous partial differential equation (2), and let \(u_{2}(x, t)\) be a solution of the linear nonhomogeneous partial differential equation (1b). Show, for any constant \(c_{1}\), that \(u(x, t)=c_{1} u_{1}(x, t)+u_{2}(x, t)\) is also a solution of the nonhomogeneous equation.

Short answer

Question: Show that the function \(u(x, t) = c_1 u_1(x, t) + u_2(x, t)\), where \(c_1\) is any constant, is a solution of the linear nonhomogeneous PDE (1b), given that (2) is a linear homogeneous PDE with a solution \(u_1(x, t)\) and (1b) is a linear nonhomogeneous PDE with a solution \(u_2(x, t)\). Answer: We have shown that after applying the linear operator \(L\) to the function \(u(x, t) = c_1 u_1(x, t) + u_2(x, t)\) and using the linearity property, we get \(L(u) = f(x, t)\), which confirms that the function \(u(x, t)\) is a solution of the linear nonhomogeneous PDE (1b).

Step-by-step solution

Write the general form of a linear homogeneous PDE and a linear nonhomogeneous PDE

A linear homogeneous PDE can be written as: $$ L(u_1) = 0, $$ where \(L\) is a linear operator, and a linear nonhomogeneous PDE can be written as: $$ L(u_2) = f(x, t), $$ where \(f(x, t)\) is a given nonhomogeneous term.

Define the function \(u(x, t)\)

We are given that for any constant \(c_1\), the function \(u(x, t)\) is defined as: $$ u(x, t) = c_1 u_1(x, t) + u_2(x, t). $$

Apply the linear operator to \(u(x, t)\)

Now, we will apply the linear operator \(L\) to the function \(u(x, t)\) and use the linearity property, which states that \(L(c_1 u_1 + u_2) = c_1 L(u_1) + L(u_2)\). $$ L(u) = L(c_1 u_1(x, t) + u_2(x, t)) = c_1 L(u_1(x, t)) + L(u_2(x, t)). $$

Substitute the given solutions of PDEs and simplify

We are given that \(L(u_1(x, t)) = 0\) and \(L(u_2(x, t)) = f(x, t)\). Substitute these given solutions into the above equation and simplify: $$ L(u) = c_1 (0) + f(x, t) = f(x, t). $$ Since \(L(u) = f(x, t)\), the function \(u(x, t) = c_1 u_1(x, t) + u_2(x, t)\) is a solution of the linear nonhomogeneous PDE (1b), as required.

Key concepts

Linear Homogeneous PDE

A linear homogeneous partial differential equation (PDE) represents a relationship where the sum of partial derivatives of a function equals zero. In mathematical terms, if we take a linear operator, denoted by \( L \), and apply it to a function \( u_1(x, t) \), we get the PDE formula:

\[ L(u_1) = 0, \]
where \( u_1(x, t) \) is the unknown function we aim to find. The term 'homogeneous' means that this equation is equal to zero, with no additional functions or constants on the right side. Because of its simplicity, linear homogeneous PDEs exhibit certain elegant properties—for instance, if you have two solutions, any linear combination of these solutions is also a solution, which leads us into discussions on the superposition principle.

Linear Nonhomogeneous PDE

On the flip side of linear PDEs, we encounter the linear nonhomogeneous PDE. This form of PDE is similar to the homogeneous case but includes a non-zero term on the right side, which can be any function of the independent variables—in our case, \( f(x, t) \). The equation is written as:

\[ L(u_2) = f(x, t), \]
where \( u_2(x, t) \) is the function to be determined. The presence of the nonhomogeneous term \( f(x, t) \) complicates the solution process, as it requires additional methods to solve, such as the method of undetermined coefficients or variation of parameters. These solutions provide insights into a wide array of physical phenomena where some external force or source is present.

Linear Operator

The linear operator, denoted \( L \), is at the core of understanding both types of PDEs. A linear operator is a rule that takes a function and primarily results in a differential operation on that function. The main characteristics of a linear operator include additivity and homogeneity, meaning the operator can be distributed over the sum of functions, and constants can be factored out. This can be summarized by:

\[ L(c_1f + c_2g) = c_1L(f) + c_2L(g), \]
for any functions \( f \) and \( g \) and constants \( c_1 \) and \( c_2 \). These properties are essential for combining solutions to PDEs and understanding the interactions between the various parts of a differential equation.

Superposition Principle

The superposition principle is a fundamental concept in linear algebra and differential equations, which allows for the construction of new solutions from known solutions. It states that if \( u_1(x, t) \) and \( u_2(x, t) \) are solutions to a linear homogeneous PDE, then any linear combination of these solutions, such as \( u(x, t) = c_1 u_1(x, t) + u_2(x, t) \) where \( c_1 \) is a constant, is also a solution to the PDE.

In our exercise, we utilized this principle to combine a specific solution \( u_2(x, t) \) of a nonhomogeneous PDE with a multiple \( c_1 u_1(x, t) \) of a homogeneous solution to get a new solution to the nonhomogeneous equation. This approach is commonly used to solve complex physical problems where different wave forms or other linear phenomena are superimposed.