# 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

The solutions of an homogeneous linear differential equation form vector space over the corresponding field.

The solutions of an inhomogeneous linear differential equation form affine space over the solutions of the corresponding homogeneous linear equation.

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

• Could you clear up on what an affine space is? My knowledge is only up to linear algebra and differential equations. – Sentient Apr 1 '17 at 1:30
• @Sentient An affine space is intuitively a vector space with the origin shifted to another point, like the solution to an inhomogeneous system of linear equations, or the solution to an inhomogeneous linear differential equation. Indeed the algebra of these two situations is identical. – Ian Apr 1 '17 at 1:55