hunt_mat said:Variations of parameters.
A nonhomogenous equation is a type of differential equation where the right-hand side of the equation is a function that is not equal to zero. This means there is an external or non-zero force acting on the system.
In a homogenous equation, the right-hand side is equal to zero, meaning there is no external force acting on the system. This makes it easier to solve compared to a nonhomogenous equation.
The general solution for a nonhomogenous equation is the sum of the complementary function and the particular integral. The complementary function is the solution to the homogenous equation, while the particular integral is a specific solution to the nonhomogenous equation.
To find the complementary function, you first solve the homogenous equation by setting the right-hand side to zero. This will give you a general solution in terms of constants. These constants can then be determined by applying any initial conditions given in the problem.
Sure, for example, the nonhomogenous equation y'' + 2y' + y = 3x + 5 can be solved by first finding the complementary function y_c = c1e^(-x) + c2xe^(-x). Then, the particular integral y_p = Ax + B can be found by substituting it into the original equation and solving for A and B. The general solution would then be y = c1e^(-x) + c2xe^(-x) + Ax + B.