Question

In Exercises \(1-12\) find the general solution. $$ y^{\prime \prime}-4 y^{\prime}+4 y=0 $$

Short answer

Question: Find the general solution of the given second-order, homogeneous, linear differential equation: $$y'' - 4y' + 4y = 0$$ Answer: The general solution of the given differential equation is $$y(x) = c_1e^{2x} + c_2xe^{2x}.$$

Step-by-step solution

Write down the given differential equation

The given differential equation is: $$y'' - 4y' + 4y = 0$$

Formulate the auxiliary (characteristic) equation

Let's write down the auxiliary equation associated with the given differential equation. Replace each derivative with the corresponding power of r: $$r^2 - 4r + 4 = 0$$

Solve the auxiliary equation

We can factor the auxiliary equation: $$(r - 2)(r - 2) = 0$$ So, we have a repeated root, \(r = 2\).

Write the general solution

Since there is a repeated root, we will use the following general solution template: $$y(x) = c_1e^{rx} + c_2xe^{rx}$$ Substitute the found value of the root, \(r = 2\), into the general solution template: $$y(x) = c_1e^{2x} + c_2xe^{2x}$$

Present the final general solution

The general solution of the given second-order, homogeneous, linear differential equation is: $$y(x) = c_1e^{2x} + c_2xe^{2x}$$

Key concepts

Characteristic Equation

When tackling second-order differential equations, the characteristic equation is a crucial concept. It's a tool to find the solution of a differential equation by transforming the differential problem into an algebraic one. In our exercise, we form the characteristic equation by replacing each derivative with a power of a variable, often denoted as r. For the equation y'' - 4y' + 4y = 0, the corresponding characteristic equation is r^2 - 4r + 4 = 0. Solving this quadratic equation is our next step towards finding the general solution to the original differential problem.

When you see a characteristic equation, remember it simplifies the process by looking for values of r that satisfy the equation. Once found, these values — known as the roots — are used to construct the solution to the differential equation. Let's see how this applies when we confront homogeneous linear differential equations next.

Homogeneous Linear Differential Equation

Our example falls under the category of homogeneous linear differential equations, which have a standard form involving derivatives of the function y(x), coefficients, and typically set equal to zero. Homogeneous refers to the absence of a non-zero function on the right side of the equation, and linear implies that the function y(x) and its derivatives appear to the first power and are not multiplied together.

Homogeneous linear differential equations, like y'' - 4y' + 4y = 0, are important in various fields from physics to economics because they describe systems that have proportional responses - if you double the input, the output doubles as well. This linearity and the lack of external forcing (homogeneity) make these equations more tractable and their solutions crucial for understanding the behavior of the system being modeled.

Second-Order Differential Equations

Our focus here is specifically on second-order differential equations, which involve the second derivative of the function, y''. These equations frequently appear in physics, representing scenarios like motion under constant acceleration. The general form is y'' + p(x)y' + q(x)y = 0, where p(x) and q(x) are functions of x.

The solutions to such equations are based on the characteristic equation's roots and can take several different forms, including exponential functions, sine and cosine functions, or a combination, depending on whether the roots are real, complex, or repeated. Our given problem features a repeated root, which leads us to a special case in the process of solving these equations.

Repeated Roots

In some instances, the characteristic equation yields repeated roots; our equation's characteristic roots are both r = 2. When this happens, the general solution to the differential equation adopts a specific form. Unlike distinct roots which yield solutions composed purely of exponential terms with different bases, a repeated root like in our case leads to a general solution that includes an additional term, c_2xe^{rx}.

This extra term accounts for the multiplicity of the root and ensures the solution space is complete. Multiplicity refers to the number of times a particular root occurs; for our example, the root 2 has a multiplicity of 2 because the characteristic equation factors into (r - 2)^2 = 0. This concept is vital as each unique solution format corresponds to different physical interpretations and behaviors in the systems modeled by the differential equation.