Overview
An equation consisting of a differential coefficient is called a differential equation. whereas a linear equation with one or more terms, consisting of the derivatives of the dependent variable with respect to one or more independent variables is called LDE.
For example-
A differential equation of the form
is known as a LDE.
The differential equation of the form dy/dx + Py = Q is the first order linear differential equation, where P and Q are the constant or function of independent variable..
P and Q, are functions of x or constants.
Few examples of LDE are-
The solution of LDE is-
Note
1.
2. If the RHS of LDE is zero for all x then it is said to be homogeneous, otherwise non-homogeneous.
Solving a linear differential equation
1. First we re-arrange the given equation to the standard form of LDE, which is dy/dx + Py = Q is the first order linear differential equation, where P and Q are the constant or function of independent variable.
2. Then we find the integrating factor
3. Then the solution of LDE is-
Solved examples
Example: Solve
Solution-
First we will convert the given equation in standard LDE form-
Where Q = sin x and P = 2/x
Now we will find the integrating factor-
Then the solution is-
Integrating by parts-
Example: Solve-
Solution-
The given equation is already in the form of standard LDE.
Now we will find the IF-
So that the solution is-
Interested in learning about similar topics? Here are a few hand-picked blogs for you!