Simple PDE: Finding the General Solution for u_{xx} + u = 6y

In summary, you don't know whether or not this is the most general solution, and you need to find a way to find out.
  • #1
quantum_smile
21
1

Homework Statement


Find the general solution of
[tex] u_{xx} + u = 6y, [/tex]
in terms of arbitrary functions.

Homework Equations



The PDE has the homogeneous solution, [tex] u(x,y)=Acos(x)+Bsin(x) [/tex].
[tex] u_{xx} + u = 6y [/tex] has the particular solution, [tex] u(x,y)=6y [/tex]

The Attempt at a Solution



Taking a superposition of the homogeneous and particular solutions, we can write that
u(x,y)=Acos(x)+Bsin(x)+6y. <--my solution

My question is, how do I know whether or not this is *the* most general solution? How do I know that I haven't missed something?
 
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  • #2
quantum_smile said:

Homework Statement


Find the general solution of
[tex] u_{xx} + u = 6y, [/tex]
in terms of arbitrary functions.


Homework Equations



The PDE has the homogeneous solution, [tex] u(x,y)=Acos(x)+Bsin(x) [/tex].
[tex] u_{xx} + u = 6y [/tex] has the particular solution, [tex] u(x,y)=6y [/tex]

The Attempt at a Solution



Taking a superposition of the homogeneous and particular solutions, we can write that
u(x,y)=Acos(x)+Bsin(x)+6y. <--my solution

My question is, how do I know whether or not this is *the* most general solution? How do I know that I haven't missed something?

You know that ##y_h## and ##y_p## are both linearly independent solutions to the equation. Therefore the sum of those two linearly independent solutions is also a linearly independent solution. Hence the most general solution (although not necessarily unique) is ##y = y_h + y_p##.

Consider a regular ODE for a moment of the form: ##a(x)y'' + b(x)y' + c(x)y = f(x)##.

Differentiating ##y## twice and plugging it into the above equation you will find the answer to be ##0 + f(x) = f(x)##.
 
  • #3
quantum_smile said:

Homework Statement


Find the general solution of
[tex] u_{xx} + u = 6y, [/tex]
in terms of arbitrary functions.

Homework Equations



The PDE has the homogeneous solution, [tex] u(x,y)=Acos(x)+Bsin(x) [/tex].
[tex] u_{xx} + u = 6y [/tex] has the particular solution, [tex] u(x,y)=6y [/tex]

The Attempt at a Solution



Taking a superposition of the homogeneous and particular solutions, we can write that
u(x,y)=Acos(x)+Bsin(x)+6y. <--my solution

My question is, how do I know whether or not this is *the* most general solution? How do I know that I haven't missed something?

You don't know that it is. And obviously it isn't because you could have$$
u(x,y) = A(y)\cos x + B(y)\sin x + 6y$$where ##A(y)## and ##B(y)## are arbitrary functions of ##y##. Is this the most general solution? It might be, but I don't know. Usually problems like this come with some boundary conditions and assumptions which guarantee a unique solution. When you have that situation, then if you come up with something that works, no matter how you found it, you know you are done because you have the only solution there is.
 

Related to Simple PDE: Finding the General Solution for u_{xx} + u = 6y

1. What is a simple PDE?

A simple PDE (partial differential equation) is a type of mathematical equation that involves multiple variables and their partial derivatives. It is used to describe relationships between these variables and their rates of change.

2. What is the general solution of a simple PDE?

The general solution of a simple PDE is a formula or set of equations that satisfies the PDE for all values of the variables. It contains an arbitrary constant or constants that can be varied to represent different solutions to the PDE.

3. How is the general solution of a simple PDE obtained?

The general solution of a simple PDE is obtained by solving the PDE using various techniques such as separation of variables, method of characteristics, or Fourier transforms. The resulting solution is then checked to ensure it satisfies the PDE, and any necessary constants are added to create the general solution.

4. What is the difference between a general solution and a particular solution of a simple PDE?

The general solution of a simple PDE represents all possible solutions to the PDE, while a particular solution is a specific solution that satisfies the PDE for given initial or boundary conditions. The general solution contains arbitrary constants, while a particular solution has specific values for these constants.

5. Why is finding the general solution of a simple PDE important?

Finding the general solution of a simple PDE is important because it allows us to obtain all possible solutions to the PDE and understand the behavior of the system described by the PDE. It also serves as a starting point for finding particular solutions that satisfy specific conditions, which are often needed in real-world applications.

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