# Why is the solution of a first-order nonhomogeneous equation equal to a particular integral and a complementary function?

I just learned today how to solve a first-order nonhomogeneous equation by breaking it down into a particular integral and a complementary function. What i'm wondering is WHY is the solution equal to these 2 things added together? I can see how each individual part of them is solved from the general term but I dont understand why when u add them together, u get the general solution to the complete differential equation?

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0% accepted rate is not a good sign here. :) – Babak S. Oct 23 '12 at 10:14

Are you asking why the general solution is composed of a homogeneous solution and a particular solution? All right, let's say we have a particular solution $y_p$ that satisfies the linear differential equation

$$\sum f_n(x)\frac{d^ny_p}{dx^n}=g(x)$$

and a corresponding homogeneous solution $y_h$ satisfying

$$\sum f_n(x)\frac{d^ny_h}{dx^n}=0$$

Now because the derivative of a sum is equal to the sum of the derivatives, we have

$$\sum f_n(x)\frac{d^n(y_p+y_h)}{dx^n}=\sum f_n(x)\frac{d^ny_p}{dx^n}+\sum f_n(x)\frac{d^ny_h}{dx^n}=g(x)+0=g(x)$$

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