Solve the following initial value problem for u as a function of t:

du/dt + (k/m) u = 0 (k and m positive constants), u(0) = u₀

b. as a separable equation.

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Rewrite the differential equation \( \frac{du}{dt} + \frac{k}{m} u = 0 \) in a form that isolates \( \frac{du}{dt} \): \[ \frac{du}{dt} = -\frac{k}{m} u \].

Recognize that this is a separable differential equation, so separate the variables \( u \) and \( t \) by dividing both sides by \( u \) and multiplying both sides by \( dt \): \[ \frac{1}{u} du = -\frac{k}{m} dt \].

Integrate both sides: \[ \int \frac{1}{u} du = \int -\frac{k}{m} dt \]. This gives \( \ln|u| = -\frac{k}{m} t + C \), where \( C \) is the constant of integration.

Solve for \( u \) by exponentiating both sides: \[ u = e^{ -\frac{k}{m} t + C } = e^{C} e^{ -\frac{k}{m} t } \]. Let \( A = e^{C} \) to simplify the expression.

Use the initial condition \( u(0) = u_0 \) to find \( A \): \[ u_0 = A e^{0} = A \]. Therefore, the solution is \[ u(t) = u_0 e^{ -\frac{k}{m} t } \].

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Separable Differential Equations

A separable differential equation can be written as the product of a function of the dependent variable and a function of the independent variable. This allows the variables to be separated on opposite sides of the equation, enabling integration with respect to each variable independently.

Initial Value Problem (IVP)

An initial value problem specifies the value of the unknown function at a particular point, providing a unique solution to a differential equation. Here, u(0) = u₀ sets the starting condition needed to determine the constant of integration after solving.

Exponential Decay Solution

The given differential equation models exponential decay, where the rate of change of u is proportional to -u. Solving the separable equation leads to a solution of the form u(t) = u₀ e^(-kt/m), showing how u decreases over time.

Solve the following initial value problem for u as a function of t:du/dt + (k/m) u = 0 (k and m positive constants), u(0) = u₀b. as a separable equation.