Differential equations - nonhomogeneous

In summary, we are given the linear transformation Ly = y'' + py' + qy. We are also given two functions y1 and y2, with Ly1 = f(x) and Ly2 = g(x). By the Superposition Principle, we know that L[c1y1 + c2y2] = c1L[y1] + c2L[y2]. Therefore, we can deduce that Ly = f(x) + g(x), where y = y1 + y2. This shows that the sum y satisfies the nonhomogeneous equation Ly = f(x) + g(x).
  • #1
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Homework Statement


Let Ly = 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 Ly 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!
 
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  • #2
The symbol L represents the particular linear transformation of this problem. Here
[tex]L \equiv \frac{d^2}{dx^2} + p\frac{d}{dx} + q[/tex]

so
[tex]Ly_1 \equiv \left(\frac{d^2}{dx^2} + p\frac{d}{dx} + q\right)y_1[/tex]

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.
 

Related to Differential equations - nonhomogeneous

1. What is a nonhomogeneous differential equation?

A nonhomogeneous differential equation is a type of differential equation where the right-hand side of the equation contains a function that is not equal to zero. This function is usually referred to as the nonhomogeneous term or the forcing function.

2. How is a nonhomogeneous differential equation different from a homogeneous one?

In a homogeneous differential equation, the right-hand side of the equation is equal to zero, meaning that there is no external input or forcing function. In contrast, a nonhomogeneous differential equation has a non-zero right-hand side, indicating the presence of an external input or forcing function.

3. What are some techniques for solving nonhomogeneous differential equations?

Some common techniques for solving nonhomogeneous differential equations include the method of undetermined coefficients, variation of parameters, and Laplace transforms. These techniques involve finding a particular solution that satisfies the nonhomogeneous term and combining it with the general solution of the corresponding homogeneous equation.

4. Can nonhomogeneous differential equations have multiple solutions?

Yes, nonhomogeneous differential equations can have multiple solutions. This is because the general solution of a nonhomogeneous differential equation includes the general solution of the corresponding homogeneous equation and a particular solution. Therefore, there can be infinite variations of particular solutions, resulting in multiple solutions for the nonhomogeneous equation.

5. What are some real-world applications of nonhomogeneous differential equations?

Nonhomogeneous differential equations have many applications in science and engineering, including modeling the motion of a damped harmonic oscillator, analyzing the growth and decay of populations, and predicting the spread of diseases. They are also used in fields such as economics, biology, and chemistry to describe various natural phenomena.

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