Boundary Value Problem for y'' + y = A + sin(2x) with y(0) = y'(\pi/2) = 2

In summary, for any real number A, there exists a unique solution to the BVP given by y = A - 1/2 sin(2x)+ (2- A)cos(x)+ 8/3 sin(x). There are no values of A for which there are infinitely many solutions, and there are no values of A for which there are no solutions.
  • #1
EvilKermit
23
0

Homework Statement



a) Solve for the BVP. Where A is a real number.
b) For what values of A does there exist a unique solutions? What is the solution?
c) For what values of A do there exist infinitely many solutions?
d) For what values of A do there exist no solutions?


Homework Equations



y'' + y = A + sin(2x)
y(0) = y'([tex]\pi[/tex]/2) = 2

The Attempt at a Solution




y = yh + yp
0 = 1+ [tex]\lambda[/tex]2
yh =c1*cos(x) + c2*cos(x)

yp = A + B*sin(2x)
y = A + B*sin(2x)
y'' = -4B*sin(2x)
A + sin(2x) = A -3B*sin(2x)
A = A, B = -1/3
yp = A - 1/3*sin(2x)


y = A - 1/3*sin(2x) + c1*cos(x) + c2*sin(x)
y' = - 2/3*sin(2x) - c1*sin(x) + c2*cos(x)
2 = A + c1
2 = 2/3 + c2
c2 = 8/3

y = A - 1/3*sin(2x) + c1*cos(x) + 8/3*sin(x)

I'm confused about answering the questions. A would be equal to all real numbers, since one could solve for c1. How can I give the solution? There is a unique solution for each value of A, which I would have to write infinite solutions. And there is no value of A when there is an inifiite amount of solutions or no values.

IF A is defined, what would the answer be?
 
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  • #2
EvilKermit said:

Homework Statement



a) Solve for the BVP. Where A is a real number.
b) For what values of A does there exist a unique solutions? What is the solution?
c) For what values of A do there exist infinitely many solutions?
d) For what values of A do there exist no solutions?


Homework Equations



y'' + y = A + sin(2x)
y(0) = y'([tex]\pi[/tex]/2) = 2

The Attempt at a Solution




y = yh + yp
0 = 1+ [tex]\lambda[/tex]2
yh =c1*cos(x) + c2*cos(x)

yp = A + B*sin(2x)
y = A + B*sin(2x)
y'' = -4B*sin(2x)
A + sin(2x) = A -3B*sin(2x)
A = A, B = -1/3
yp = A - 1/3*sin(2x)


y = A - 1/3*sin(2x) + c1*cos(x) + c2*sin(x)
y' = - 2/3*sin(2x) - c1*sin(x) + c2*cos(x)
2 = A + c1
So c1= 2-A

2 = 2/3 + c2
c2 = 8/3

y = A - 1/3*sin(2x) + c1*cos(x) + 8/3*sin(x)
y= A- 1/2 sin(2x)+ (2- A)cos(x)+ 8/3 sin(x)

I'm confused about answering the questions. A would be equal to all real numbers, since one could solve for c1. How can I give the solution? There is a unique solution for each value of A, which I would have to write infinite solutions. And there is no value of A when there is an inifiite amount of solutions or no values.

IF A is defined, what would the answer be?
 
  • #3
So:
y= A- 1/2 sin(2x)+ (2- A)cos(x)+ 8/3 sin(x)

c) For what values of A do there exist infinitely many solutions?
d) For what values of A do there exist no solutions?

Would c and d then be no values of A have infinitely many solutions or nonexistent solution?
 
  • #4
Yes.
 

What is a boundary value problem?

A boundary value problem is a type of mathematical problem in which the solution is sought for a set of differential equations subject to a set of boundary conditions. Boundary value problems are important in fields such as physics, engineering, and mathematics.

What are the types of boundary value problems?

There are two main types of boundary value problems: Dirichlet boundary value problems, where the boundary conditions specify the value of the solution at the boundary, and Neumann boundary value problems, where the boundary conditions specify the derivative of the solution at the boundary.

How are boundary value problems solved?

Boundary value problems are typically solved using numerical methods, such as finite difference methods, finite element methods, or spectral methods. These methods discretize the problem into a set of algebraic equations, which can then be solved using a computer.

Why are boundary value problems important?

Boundary value problems are important because they allow us to model and understand physical systems, such as heat flow, fluid dynamics, and electromagnetic fields. They also have many practical applications, such as in engineering design and optimization.

What are some real-world examples of boundary value problems?

Some examples of boundary value problems include calculating the temperature distribution in a metal rod subjected to different thermal boundary conditions, determining the flow of water around a ship hull given different boundary conditions, and finding the electric potential in a circuit with different boundary conditions.

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