91Ó°ÊÓ

Differential equations are equations that involve functions and their derivatives. In the context of circuits, these equations are vital in describing how current and voltage change over time. For an inductor circuit, which is what we're focusing on, the relationship between voltage and current is captured by the equation\( v_L(t) = L \frac{di_L(t)}{dt} \). Here, \( L \) represents the inductance, and \( v_L(t) \) and \( i_L(t) \) are the voltage and current, respectively.
In our specific problem, because the inductance \( L = 1 \) Henry, the differential equation simplifies to \( \frac{di_L(t)}{dt} = v_L(t) \). Given that \( v_L(t) = t \exp(-t) \), we substitute this into the equation to form \( \frac{di_L(t)}{dt} = t \exp(-t) \). This establishes a first-order linear differential equation that can be solved to find \( i_L(t) \).
Solving such equations often involves integration or other techniques like Laplace transforms. Understanding how to set up and solve these differential equations is crucial for anyone delving into electronics and circuit design.

Integration by parts is a technique used to solve integrals that involve products of functions. This method is particularly useful when dealing with integrals like \( \int t \exp(-t) \, dt \). The formula for integration by parts is given as:

• \( \int u \, dv = uv - \int v \, du \)

To use this method, we select parts of the integral to be \( u \) and \( dv \). In our calculus problem, let \( u = t \) which makes \( du = dt \), and \( dv = \exp(-t) \, dt \) which implies \( v = -\exp(-t) \).
Using integration by parts leads to the integral\( \int t \exp(-t) \, dt = -t \exp(-t) - \int -\exp(-t) \, dt \). Solving this further, we find:\( i_L(t) = -t \exp(-t) - \exp(-t) + C \), where \( C \) is the constant determined by initial conditions.
This integral process is crucial in solving for current, \( i_L(t) \), and confirms how integration by parts can transform complex calculus problems into solvable components.

MATLAB is a powerful tool frequently used in engineering to solve and visualize complex problems. In electrical engineering, MATLAB can simulate circuits and solve differential equations, helping verify analytical solutions. In our exercise, MATLAB helps us graph the functions \( v_L(t) = t \exp(-t) \) and \( i_L(t) = -t \exp(-t) - \exp(-t) + 1 \).
To verify our solution using MATLAB, follow these basic steps:

• Define the functions for \( v_L(t) \) and \( i_L(t) \).
• Use `fplot` to plot both functions over the desired interval, \( 0 \leq t \leq 10 \).
• Use `hold on` to ensure both plots appear on the same graph.

These steps enable us to visually compare \( v_L(t) \) and \( i_L(t) \), ensuring that the solutions obtained analytically match their graphical representation.
By integrating MATLAB into electrical engineering studies, students can enhance their understanding of circuit behavior and confirm the results they calculate by hand.