# Necessity of Homogeneous Solution When Solving Nonhomgeneous Differential Equation

By the superposition principal for linear differential equations, if $y_p$ is the particular solution to the differential equation., and $y_h$ is the homogeneous solution, then $y_p + y_h$ also a solution to the differential equation.

Why is it necessary to include $y_h$ in the solution?

What is the geometric intuition behind it? In linear algebra, the homogeneous solution represented the solution set going through the origin, and the particular solution was a vector that shifted the solution set to the 'correct' location.

• If you can stretch your intuition to encompass the notion that "linearly independent functions" are just like "linearly independent vectors" in linear algebra, the situation is exactly the same. – Ian Apr 1 '17 at 1:57

Then, you must write $y_p + y_h$ to consider all the solutions.