Problem 122 - Calculus 3

Question

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. $$y^{\prime \prime}-3 y+2 y=\cos (t)$$

Step-by-Step Solution

Answer

The differential equation is second-order, linear, and nonhomogeneous.

Step 1: Identify the Order

The order of a differential equation is the highest derivative present in the equation. In the given differential equation \(y'' - 3y + 2y = \cos(t)\), the highest derivative is \(y''\), which is the second derivative. Therefore, the order of this differential equation is 2.

Step 2: Check for Linearity

A differential equation is linear if it can be written in the form: \(a_n(t) y^{(n)} + a_{n-1}(t) y^{(n-1)} + \,\ldots\, + a_1(t) y' + a_0(t) y = g(t)\), where all terms involving \(y\) or its derivatives are of the first degree and the coefficients are functions of \(t\) or constants. In the given equation \(y'' - 3y + 2y = \cos(t)\), the terms are linear, and the equation can be arranged as \(y'' - y = \cos(t)\). Therefore, it is a linear differential equation.

Step 3: Determine Homogeneity

A linear differential equation is homogeneous if the term on the right side is zero, i.e., \(g(t) = 0\). In this case, the right side of the equation is \(\cos(t)\), which is not zero. Therefore, the differential equation \(y'' - y = \cos(t)\) is nonhomogeneous.

Key Concepts

Differential equation classification

Differential equations, which are equations that relate a function to its derivatives, can be classified based on several criteria. One of the primary ways to classify them is by their order, which is determined by the highest derivative present in the equation. In the exercise provided, the highest derivative is the second derivative, denoted as \( y'' \). Therefore, this is a second-order differential equation.

Another critical classification is based on linearity. A differential equation is linear if each term involving the function or its derivatives is to the first power, and if the coefficients of these terms are functions of the independent variable or constants. If any term involves the square (or higher power) of the function or its derivatives, then the equation is nonlinear.

Lastly, differential equations can be classified as either homogeneous or nonhomogeneous, based on whether the non-derivative part (the term on the non-left hand side) equals zero.
Understanding these classifications is essential to determine the techniques that can be applied to find a solution.

Linear differential equations

A linear differential equation is characterized by having all terms involving the unknown function or its derivatives raised only to the first power. Furthermore, there are no products of the function and its derivatives. The coefficients of these terms should only be functions of the independent variable or constants. For example, the linear equation in the exercise is \( y'' - y = \cos(t) \).

In general, a linear second-order differential equation can be expressed in the standard form: \[ a(t)y'' + b(t)y' + c(t)y = g(t) \] where \( a(t) \), \( b(t) \), and \( c(t) \) are functions of the independent variable \( t \), and \( g(t) \) represents any forcing term on the right side of the equation.

Linear differential equations are significant because they can often be solved using systematic methods, such as undetermined coefficients or variation of parameters, which are typically not applicable to nonlinear equations.

Nonhomogeneous differential equations

A differential equation is considered nonhomogeneous if it includes a term that is not dependent on the function or its derivatives, typically presented on the right-hand side of the equation. In the case of the equation from the exercise, \( y'' - y = \cos(t) \), \( \cos(t) \) serves as this nonhomogeneous term.

In mathematical terms, a nonhomogeneous differential equation can be written as:\[ a_n(t)y^{(n)} + a_{n-1}(t)y^{(n-1)} + \,\ldots\, + a_1(t)y' + a_0(t)y = g(t) \]where \( g(t) eq 0 \). This distinguishes nonhomogeneous from homogeneous equations, where this term would be zero, \( g(t) = 0 \).

Solving nonhomogeneous differential equations usually involves finding a particular solution that satisfies the entire equation and a complementary solution solving the associated homogeneous equation. Together, these solutions provide the general solution for nonhomogeneous differential equations.