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Boundary Conditions Calculator

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Example
Created on 2024-06-20Asked by Daniel Rivera (Solvelet student)
Solve the differential equation d2ydx2=−y \frac{d^2y}{dx^2} = -y with boundary conditions y(0)=0 y(0) = 0 and y(π)=0 y(\pi) = 0 .

Solution

Step 1: Identify the differential equation and boundary conditions: d2ydx2=−y,y(0)=0,y(π)=0. \frac{d^2y}{dx^2} = -y, \quad y(0) = 0, \quad y(\pi) = 0. Step 2: Solve the characteristic equation r2+1=0 r^2 + 1 = 0 : r=±i. r = \pm i. Step 3: Write the general solution of the differential equation: y(x)=Acos⁡(x)+Bsin⁡(x). y(x) = A \cos(x) + B \sin(x). Step 4: Apply the boundary condition y(0)=0 y(0) = 0 : y(0)=Acos⁡(0)+Bsin⁡(0)=A=0. y(0) = A \cos(0) + B \sin(0) = A = 0. Step 5: Apply the boundary condition y(π)=0 y(\pi) = 0 : y(π)=Bsin⁡(π)=0. y(\pi) = B \sin(\pi) = 0. Step 6: Conclusion. The only solution that satisfies both boundary conditions is y(x)=0 y(x) = 0 . Solved on Solvelet with Basic AI Model
Some of the related questions asked by Scarlett Torres on Solvelet
1. Apply the boundary condition u(0)=0 u(0) = 0 to the heat equation ∂u∂t=k⋅∂2u∂x2 \frac{\partial u}{\partial t} = k \cdot \frac{\partial^2 u}{\partial x^2} 2. Determine the boundary condition necessary to uniquely specify the solution of the wave equation ∂2u∂t2=c2∂2u∂x2 \frac{\partial^2u}{\partial t^2} = c^2 \frac{\partial^2u}{\partial x^2} .,
DefinitionIn the field of mathematics and physics, boundary conditions which are also known as initial conditions are prerequisites that the solutions of differential equations have to satisfy in order to be unique within a certain class of solutions. They delimit the way the solution behaves at the domain boundaries or interface, and contain typically data on the function position and derivatives, or both. In physics,engineering and applied mathematics, boundary condition s are the extra conditions, a DEQ or PDE solution needs to provide to correctly solve the problem. For example, in a heat conduction problem, boundary conditions tell us what the temperature of a material is on the boundaries such as constant temperature or heat flux.
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