Linear Differential Equations
Define Linear Differential Equations.
A linear differential equation refers to an equation that involves a dependent variable and its derivatives. In this type of equation, the linearity property is maintained with respect to the dependent variable and its derivatives.
What is the General Solution of First Order Linear Differential Equations?
General solution of first order linear differential equations is y × I.F = ∫Q × I.F dt + C, where C is the integration constant and I.F is the integrating factor.
How do you Know if its a Linear Differential Equation?
A differential equation is linear if all its terms involving the unknown function and its derivatives are either constants or multiples of the unknown function and its derivatives raised to the power of 1.
What is Integrating Factor?
Integrating factor is a function which is selected in order to solve the given differential equation. If the differential equation is of the form dy/dx + P(x)y = Q(x) then the integrating factor is given by I.F = e∫P dx.
What is Particular Integral?
Particular integral refers to a specific solution that satisfies the differential equation when combined with the complementary function.
How to Solve Second Order Linear Differential Equations?
Steps to solve a Second Order Linear Differential Equations are:
- Find the characteristics equation.
- Solve the Characteristics equation.
- Apply the initial or boundary conditions.
What are the Differences between Linear and Non-Linear Differential Equation?
Linear differential equations have terms with variables and their derivatives raised to the power of 1, while nonlinear differential equations have terms with variables and their derivatives raised to powers greater than 1.
Linear Differential Equations
Linear Differential Equations are differential equations where the unknown function and its derivatives appear linearly. In other words, the equation is a linear combination of the function and its derivative, with constant coefficients. Such types of equations have solutions that can be expressed as a sum of particular and homogeneous solutions.
In this article, we will discuss all things linear differential equations, including their order, type, solutions, and applications in various fields of mathematics and science.
Table of Content
- What are Linear Differential Equations?
- Examples of Linear Differential Equations
- Order of Linear Differential Equations
- Formula for General Solution of Linear Differential Equations
- Formula for First-Order Linear ODE
- Formula for Second-Order Linear ODE
- First Order Linear Differential Equations
- Examples of Linear Differential Equations
- How to Solve First-Order Linear Differential Equation?
- Second-Order Linear Differential Equation
- How to Solve Second Order Linear Differential Equation?
- A. For Homogeneous Second Order Linear Differential Equation:
- B. For Non-Homogeneous Second Order Differential Equation:
- Linear Differential Equation Formula
- Non-Linear Differential Equation
- Linear vs Non-Linear Differential Equation
- Homogeneous and Non Homogeneous Linear Differential Equations
- Conclusion
- Examples on Linear Differential Equation
- Practice Questions on Linear Differential Equation