Differential equations - nonhomogeneous

[accountkiller]

Homework Statement

Let L_y = y'' + py' + qy. Suppose y₁ and y₂ are functions such that Ly₁ = f(x) and Ly₂ = g(x). Show that the sum y = y₁ + y₂ satisfies the nonhomogeneous equation Ly = f(x) + g(x).

Homework Equations

Superposition Principle: L[c₁y₁ + c₂y₂] = c₁L[y₁] + c₂L[y₂]

The Attempt at a Solution

Even though this problem is in the book problems, I don't even see anything with an L_y in the section paragraphs. All we had from lecture is the above superposition principle. I don't even know how to start - I'd really appreciate any help!

 

[Mark44]

The symbol L represents the particular linear transformation of this problem. Here
$$L \equiv \frac{d^2}{dx^2} + p\frac{d}{dx} + q$$

so
$$Ly_1 \equiv \left(\frac{d^2}{dx^2} + p\frac{d}{dx} + q\right)y_1$$

or Ly₁ = y₁'' + py₁' + qy₁

You need to show that Ly = f(x) + g(x), where y = y₁ + y₂

What is Ly here? Start by replacing y.