2.2 Homogeneous and Non-homogeneous Solutions

Linear differential equations come in two flavors: homogeneous and non-homogeneous. Homogeneous equations have zero on the right side, while non-homogeneous ones have a non-zero function. This distinction affects how we solve them and what their solutions look like.

For homogeneous equations, we find a fundamental set of solutions. Non-homogeneous equations require an extra step: finding a particular solution. Combining these gives us the complete solution, which we can tailor to specific conditions in real-world problems.

Homogeneous vs Non-homogeneous Equations

Linear Differential Equations

• A linear differential equation involves derivatives of an unknown function and has the form $a_n(x)y^((n))(x) + ... + a_1(x)y'(x) + a_0(x)y(x) = f(x)$, where a_i(x) and f(x) are continuous functions on an interval I
• The order of the linear differential equation is determined by the highest derivative present in the equation (first-order, second-order, etc.)
• Linear differential equations are used to model various phenomena in physics, engineering, and other fields (population growth, electrical circuits, mechanical systems)

Homogeneous and Non-homogeneous Equations

• A linear differential equation is homogeneous if f(x) = 0 for all x in the interval I
  • The right-hand side of the equation is equal to zero
  • Example: $y'' + 4y' + 4y = 0$
• A linear differential equation is non-homogeneous if f(x) ≠ 0 for at least one x in the interval I
  • The right-hand side of the equation is a non-zero function
  • Example: $y'' + 4y' + 4y = e^x$
• The presence or absence of the non-zero function f(x) on the right-hand side determines whether the equation is homogeneous or non-homogeneous, respectively

General Solutions for Homogeneous Equations

Fundamental Set of Solutions

• The general solution of a homogeneous linear differential equation is a linear combination of linearly independent solutions, called the fundamental set of solutions
• For an n-th order homogeneous linear differential equation, the general solution is $y(x) = c_1y_1(x) + c_2y_2(x) + ... + c_ny_n(x)$, where y_1(x), y_2(x), ..., y_n(x) form a fundamental set of solutions and c_1, c_2, ..., c_n are arbitrary constants
• The number of linearly independent solutions in the fundamental set is equal to the order of the differential equation
  • A second-order equation will have two linearly independent solutions in its fundamental set
  • A third-order equation will have three linearly independent solutions in its fundamental set

Methods for Finding the Fundamental Set

• The characteristic equation method is used for equations with constant coefficients
  • Substitute y(x) = e^(rx) into the differential equation and solve the resulting algebraic equation for r to obtain the characteristic roots
  • Example: For $y'' - 5y' + 6y = 0$, the characteristic equation is $r^2 - 5r + 6 = 0$, which gives roots r = 2 and r = 3
• The power series method is used for equations with variable coefficients
  • Assume a solution of the form $y(x) = ∑(n=0 to ∞) a_n(x-x_0)^n$ and determine the coefficients a_n by substituting the series into the differential equation and equating coefficients
  • Example: For $xy'' + y' - xy = 0$, the power series solution is $y(x) = c_1x + c_2(x + x^2)$

Particular Solutions for Non-homogeneous Equations

Method of Undetermined Coefficients

• The method of undetermined coefficients is used when f(x) is a polynomial, exponential, sine, cosine, or a combination of these functions
• Assume a particular solution with unknown coefficients based on the form of f(x) and its derivatives
  • If f(x) is a polynomial of degree n, assume y_p(x) is a polynomial of degree n with unknown coefficients
  • If f(x) is an exponential e^(ax), assume y_p(x) = Ae^(ax) with unknown coefficient A
• Substitute the assumed solution into the differential equation and equate coefficients to determine the values of the unknown coefficients
  • Example: For $y'' - 3y' + 2y = 4e^x$, assume y_p(x) = Ae^x and solve for A

Method of Variation of Parameters

• The method of variation of parameters is a general method for finding particular solutions, especially when f(x) is not of a form suitable for the method of undetermined coefficients
• Find the fundamental set of solutions {y_1(x), y_2(x), ..., y_n(x)} for the corresponding homogeneous equation
• Assume a particular solution of the form $y_p(x) = u_1(x)y_1(x) + u_2(x)y_2(x) + ... + u_n(x)y_n(x)$, where u_1(x), u_2(x), ..., u_n(x) are unknown functions to be determined
• Substitute y_p(x) into the differential equation and solve the resulting system of equations to find u_1(x), u_2(x), ..., u_n(x)
  • Example: For $y'' + y = \sec x$, the fundamental set is {sin x, cos x}, and the particular solution is $y_p(x) = \frac{1}{2}x\sin x$

Complete Solutions for Non-homogeneous Equations

Combining Homogeneous and Particular Solutions

• The complete solution to a non-homogeneous linear differential equation is the sum of the general solution to the corresponding homogeneous equation and a particular solution to the non-homogeneous equation
• The complete solution is given by $y(x) = y_h(x) + y_p(x)$, where y_h(x) is the general solution to the homogeneous equation and y_p(x) is a particular solution to the non-homogeneous equation
  • Example: For $y'' + 4y = 3\sin 2x$, the complete solution is $y(x) = c_1\cos 2x + c_2\sin 2x + \frac{3}{8}x\cos 2x$

Applying Initial or Boundary Conditions

• The arbitrary constants in the general solution y_h(x) are determined by applying initial or boundary conditions specific to the problem
• Initial conditions specify the values of the function and/or its derivatives at a particular point (usually x = 0)
  • Example: y(0) = 1 and y'(0) = 0
• Boundary conditions specify the values of the function and/or its derivatives at two or more points
  • Example: y(0) = 0 and y(π) = 0
• The particular solution y_p(x) is found using one of the methods mentioned earlier, such as the method of undetermined coefficients or variation of parameters
• The complete solution satisfies both the differential equation and the given initial or boundary conditions, providing a unique solution to the problem