7.2 Nonhomogeneous Linear Equations

Nonhomogeneous linear equations are a key part of differential equations. They combine solutions from homogeneous equations with specific solutions for nonhomogeneous terms, giving us a complete picture of the system's behavior.

We use methods like undetermined coefficients and variation of parameters to solve these equations. These techniques help us tackle real-world problems in physics, engineering, and other fields where systems change over time.

Nonhomogeneous Linear Equations

General solution of nonhomogeneous equations

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• Sum of complementary solution (solution to associated homogeneous equation) and particular solution (specific solution accounting for nonhomogeneous term)
• Expressed as $y(t) = [y_c(t)](https://www.fiveableKeyTerm:y_c(t)) + [y_p(t)](https://www.fiveableKeyTerm:y_p(t))$
  • $y_c(t)$ represents complementary solution
  • $y_p(t)$ represents particular solution
• Homogeneous solutions contain only complementary solution while nonhomogeneous solutions (also known as inhomogeneous equations) include both complementary and particular solutions ($y_c(t)$ and $y_p(t)$)

Method of undetermined coefficients

• Finds particular solution when nonhomogeneous term is polynomial, exponential, sine, cosine, or combination of these
• Steps:
  1. Determine form of particular solution based on nonhomogeneous term
     • Polynomial of degree $n$ nonhomogeneous term leads to polynomial of degree $n$ particular solution with undetermined coefficients
     • Exponential function $e^{at}$ nonhomogeneous term leads to $Ae^{at}$ particular solution with undetermined coefficient $A$
     • Sine or cosine function ($\sin(bt)$ or $\cos(bt)$) nonhomogeneous term leads to $A\sin(bt) + B\cos(bt)$ particular solution with undetermined coefficients $A$ and $B$
  2. If nonhomogeneous term combines above, particular solution is sum of individual particular solutions
  3. Substitute assumed particular solution into nonhomogeneous differential equation and solve for undetermined coefficients
  4. Add particular solution to complementary solution to obtain general solution

Variation of parameters technique

• More general approach to finding particular solutions when method of undetermined coefficients not applicable
• Steps:
  1. Find complementary solution $y_c(t) = c_1y_1(t) + c_2y_2(t)$ with linearly independent solutions $y_1(t)$ and $y_2(t)$ to associated homogeneous equation
  2. Replace constants $c_1$ and $c_2$ with functions $u_1(t)$ and $u_2(t)$
  3. Assume $u_1'(t)y_1(t) + u_2'(t)y_2(t) = 0$ to obtain equation relating $u_1'(t)$ and $u_2'(t)$
  4. Substitute assumed particular solution $y_p(t) = u_1(t)y_1(t) + u_2(t)y_2(t)$ into nonhomogeneous differential equation to obtain another equation relating $u_1'(t)$ and $u_2'(t)$
  5. Solve system of equations for $u_1'(t)$ and $u_2'(t)$, then integrate to find $u_1(t)$ and $u_2(t)$
  6. Substitute $u_1(t)$ and $u_2(t)$ into assumed particular solution to obtain actual particular solution
  7. Add particular solution to complementary solution to obtain general solution

Applications in Differential Equations

• Nonhomogeneous linear equations are a type of differential equations solved using linear operators
• Solutions to these equations can be used to solve initial value problems and boundary value problems
• Initial value problems involve finding a solution that satisfies given initial conditions
• Boundary value problems involve finding a solution that satisfies conditions at different points in the domain