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An LR circuit consists of an inductor (L) and a resistor (R) connected in series with each other. These circuits are important in electrical engineering due to their ability to filter signals and manage current flow. The inductor stores energy in a magnetic field when electrical current passes through it, while the resistor dissipates energy in the form of heat.

When analyzing an LR circuit, it's essential to understand that the behavior of current over time can be modeled using differential equations. The key components that dictate the circuit's response are the values of inductance (measured in henries) and resistance (measured in ohms). Together, these influence how fast the current builds up or decays over time.

• Inductor (L): stores energy and opposes changes in current.
• Resistor (R): controls the rate of current flow and energy dissipation.

The fundamental equation for an LR circuit is: \( L \frac{di(t)}{dt} + Ri(t) = E(t) \), where \( E(t) \) is the electromotive force applied to the circuit. By solving this equation, you can determine the current \( i(t) \) at any time \( t \).

A piecewise function is a mathematical expression defined by different formulas or rules over different intervals of its domain. In the context of this problem, the electromotive force \( E(t) \) is a piecewise function. It is defined as 120 volts for the interval \(0 \leq t \leq 20\) and 0 volts for \(t > 20\).

Piecewise functions are useful when modeling situations where a system behaves differently at different times or under different conditions. In our problem, this behavior can be due to switching the power supply on or off at specified times.

• Defined for specific domains: Different behaviors based on the value of \(t\).
• Practical in real-world applications: Reflects situations where output changes suddenly.

Piecewise functions allow differential equations to consider these changes, enabling accurate modeling and predictions.

In differential equations, an initial condition is a given value that specifies the state of a function or its derivative at a particular time. These conditions are essential for determining the unique solution to a differential equation.

For the LR circuit problem, we have an initial condition of \(i(0) = 0\). This tells us that the current at time \(t = 0\) is zero amperes. Without this initial condition, the solution to the differential equation would not be unique, as there would be an arbitrary constant left undetermined.

• Defines the initial state: Specifies current flow at \(t = 0\).
• Ensures a unique solution: Essential for finding precise results.

By applying initial conditions, we can solve for unknown constants and ensure that our solution correctly models the system's dynamics over time.

When solving linear differential equations like those seen in LR circuits, it often involves finding both the homogeneous and particular solutions.

The homogeneous solution is the solution to the differential equation when the external force or input \(E(t) = 0\). This means we're only considering the circuit's response due to its properties without any driving source. In our case, the homogeneous solution \( i_h(t) \) is formulated as \( C e^{-\frac{t}{10}} \).

• Homogeneous solution: Models natural circuit response without external force.
• Exponential decay: Characterizes how current naturally dissipates over time.

The particular solution, on the other hand, satisfies the non-homogeneous equation, where an external force or function \(E(t)\) is applied. Here, the particular solution \( i_p(t) \) is a constant value of 60, reflecting the steady state reached due to sustained external input.

• Particular solution: Captures the circuit's response to an external force.
• Steady state: Constant solution arising from continuous external force over time.

By combining these, we obtain the general solution which reflects both the inherent behavior of the circuit and its response to external influences.