First-order differential equations are mathematical equations that describe how one quantity depends on another quantity. They can be used to describe many phenomena in nature, such as population growth, battery discharge, or fluid flow.
A first-order differential equation can be written as: dy/dx = f(x, y), where x and y are independent and dependent variables respectively, and f(x, y) is a function that describes the relation between x and y.
Differential equations (Diff. Eq.) with separable variables 1
Form :
Example:
Solve Diff. Eq.
Solution: The equation
We obtain:
We can summarize all the solutions by the single notation:
Linear differential equations of order 1
Definition:
Linear: a function and its derivative occur separately
Form :
“With constant coefficients” means that
Second member: function
Equation without 2nd member
Let
All the solutions of
Equation with 2nd member
Search for a particular solution: Constant variation method
Let
We set:
All the solutions of
Where the function
Let Diff. Eq :
To determine the particular solution, we apply the method of variation of the constant:
Diff. Eq. of order 1 with constant coefficients
We can look for a particular solution of the same type as the 2nd member
This method works for functions such as polynomials, sinusoids or exponentials.
Example
Let Diff. Eq. :
We note
To determine the particular solution, we seek a solution of the form:
We calculate the derivative:
We get the following system:
Differential equations (Diff. Eq.) with separable variables 2
Can be written:
Resolution
It is necessary to primitiveize the 2 members of the equality:
The solutions
If we can invert
To determine the particular solution of Diff. Eq. checking
Find the general solution then determine
Or, integrate the equation in separate form from the initial condition (2):
The solution of
Example:
Determine the solution of Diff. Eq. :
We integrate:
Linear differential equations of order 1
Form :
If
Principle of resolution:
1) Solve Diff. Eq. without 2nd member: homogeneous equation
After solving, we obtain the general solution in the form:
Example:
2 methods:
– By intuition, often the case when the 2nd member is constant or a simple polynomial. We seek the particular solution by identification.
– Using the constant variation method:
We replace the constant \lambda by a function
Solve:
Tagged: differential equations; first order; flow of a fluid; resolution