Let’s start by discussing a homogeneous differential equation. Differential equations have a standard form and can be written as follows:
Ay” + By’ + Cy = 0
In terms of notation, y’ = dy/dt, etc.
Note this can be expanded to higher order differential equations. For example,
Ay”’ + etc.
The above equation (Ay” + By’ + Cy = 0) is a homogeneous differential equation because the right hand side (RHS) is zero when written in the standard form.
The solution to this differential equation will not have a particular solution because it is a homogeneous differential equation.
If you had a differential equation that looked as follows:
Ay” + By’ + Cy = g
Then the RHS is non-zero and thus this is a non-homogeneous differential equation and we need to calculate the particular solution. Note that the particular solution to this differential equation is not the entire solution, however. You would need the homogeneous solution and the particular solution to calculate the general solution to this.
y_general_solution = y_homogeneous_solution + y_particular_solution
Happy solving!
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