Differential Equations: Linear Equations

Differential Equations

Linear Equations

• A linear differential equation is a differential equation in which the derivative of the unknown function is proportional to the function itself.
• General form: dy/dx + P(x)y = Q(x)
• P(x) and Q(x) are functions of x
• Properties:
  • Linear homogeneity: If y1 and y2 are solutions, then c1y1 + c2y2 is also a solution for any constants c1 and c2.
  • Superposition principle: The general solution is the sum of the homogeneous solution and the particular solution.

First Order Equations

• A first-order differential equation is a differential equation that involves only the first derivative of the unknown function.
• General form: dy/dx = f(x, y)
• Types:
  • Separable variables: dy/dx = f(x)g(y), can be solved by separating variables and integrating.
  • Linear: dy/dx + P(x)y = Q(x), can be solved using an integrating factor.
  • Exact: M(x, y)dx + N(x, y)dy = 0, can be solved by finding an implicit solution.
• Solution methods:
  • Direct integration: y = ∫f(x)dx + C
  • Separation of variables: ∫dy/g(y) = ∫f(x)dx + C

Differential Equations

Linear Equations

• General form of linear differential equation: dy/dx + P(x)y = Q(x)
• P(x) and Q(x) are functions of x
• Properties of linear differential equations:
  • Linear homogeneity: If y1 and y2 are solutions, then c1y1 + c2y2 is also a solution for any constants c1 and c2
  • Superposition principle: The general solution is the sum of the homogeneous solution and the particular solution

First Order Equations

• General form of first-order differential equation: dy/dx = f(x, y)
• Types of first-order differential equations:
  • Separable variables: dy/dx = f(x)g(y), can be solved by separating variables and integrating
  • Linear: dy/dx + P(x)y = Q(x), can be solved using an integrating factor
  • Exact: M(x, y)dx + N(x, y)dy = 0, can be solved by finding an implicit solution
• Solution methods:
  • Direct integration: y = ∫f(x)dx + C
  • Separation of variables: ∫dy/g(y) = ∫f(x)dx + C

Learn about linear differential equations, their general form, and properties such as linear homogeneity and superposition principle.