Question

The general solution of a nonhomogeneous differential equation is the sum of the homogeneous solution and the specific solution for the nonhomogeneous case. Explain why the general solution can include the homogeneous solution, but the homogeneous solution by itself is not a solution.

Answer 1

The general solution of a nonhomogeneous differential equation is the sum of the homogeneous solution and the specific solution for the nonhomogeneous case.

Step-by-step explanation:

The general solution of a nonhomogeneous differential equation is the sum of the homogeneous solution and the specific solution for the nonhomogeneous case. Let's say we have a differential equation of the form dy/dx = f(x), where f(x) is a function. The homogeneous solution is the solution to the equation dy/dx = 0, which means the derivative of y with respect to x is zero. The solution to this equation is a constant. However, this constant by itself does not satisfy the original nonhomogeneous equation, dy/dx = f(x). Therefore, the general solution includes the homogeneous solution plus a specific solution that satisfies the nonhomogeneous equation.