Separation of Variables Calculator

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Example

Created on 2024-06-20Asked by Michael Jones (Solvelet student)

Solve the partial differential equation

\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}

using separation of variables.

Solution

To solve the partial differential equation

\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}

using separation of variables, let

u(x, t) = X(x)T(t)

: 1. **Substitute

u(x, t) = X(x)T(t)

into the PDE:**

X(x) \frac{dT(t)}{dt} = k T(t) \frac{d^2X(x)}{dx^2}.

2. **Separate variables:**

\frac{1}{k} \frac{1}{T(t)} \frac{dT(t)}{dt} = \frac{1}{X(x)} \frac{d^2X(x)}{dx^2} = -\lambda.

This gives us two ordinary differential equations:

\frac{dT(t)}{dt} + \lambda k T(t) = 0,

\frac{d^2X(x)}{dx^2} + \lambda X(x) = 0.

3. **Solve the temporal part:**

T(t) = T_0 e^{-\lambda k t}.

4. **Solve the spatial part:**

X(x) = A \cos(\sqrt{\lambda} x) + B \sin(\sqrt{\lambda} x).

5. **Combine the solutions:**

u(x, t) = \left[A \cos(\sqrt{\lambda} x) + B \sin(\sqrt{\lambda} x)\right] e^{-\lambda k t}.

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Harper Brown on Solvelet

1. Solve the initial value problem

y' = y^2

y(0) = 1

using separation of variables.2. Apply separation of variables to solve the differential equation

y' = xy

with initial condition

y(0) = 1

DefinitionThe method of solving partial differential equations using the separation of variables concept is known as. This is done by assuming that every variable can be decomposed and expressed as a product of single-variable functions. Example: For ∂t∂u=k∂x2∂2u, suppose u(x,t)=X(x)T(t), then we have 2 ordinary differential equations for X and T respectively.

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