Boundary Value Problems Calculator

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Example

Created on 2024-06-20Asked by Mason Sanchez (Solvelet student)

Solve the boundary value problem

y'' + y = 0

with

y(0) = 2

and

y(\pi) = -2

Solution

Step 1: Identify the differential equation and boundary conditions:

y'' + y = 0, \quad y(0) = 2, \quad y(\pi) = -2.

Step 2: Solve the characteristic equation

r^2 + 1 = 0

r = \pm i.

Step 3: Write the general solution of the differential equation:

y(x) = A \cos(x) + B \sin(x).

Step 4: Apply the boundary condition

y(0) = 2

y(0) = A \cos(0) + B \sin(0) = A = 2.

Step 5: Apply the boundary condition

y(\pi) = -2

y(\pi) = 2 \cos(\pi) + B \sin(\pi) = -2.

-2 + 0 = -2.

Step 6: Conclusion. The solution is:

y(x) = 2 \cos(x).

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Logan Torres on Solvelet

1. Solve the boundary value problem

y'' + y = 0

y(0) = 0

y(\pi) = 2

2. Find the solution to the boundary value problem

u''(x) = -u(x)

u(0) = 1

u(\pi) = -1

DefinitionA typical example is when you are solving a differential equation subject to certain conditions - a boundary value problem (BVP) They occur in many different contexts in physics, engineering and in mathematical models with spatial dependence. Boundary value problems are problems that require the solution to satisfy a differential equation within a domain of the function and satisfy conditions at the boundaries. They are crucial in the perception of a partial differential equation and therefore the analysis of physical systems with spatial variation. Example: The heat equation is a boundary value problem for the temperature distribution over a rod given heat is being transferred at its ends and its ends are kept at fixed temperatures.

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