Problem 27 - Advanced Engineering Mathematics

Question

Solve the given initial-value problem. Give the largest interval \(I\) over which the solution is defined. $$ L \frac{d i}{d t}+R i=E ; \quad i(0)=i_{0}, L, R, E, \text { and } i_{0} \text { constants } $$

Step-by-Step Solution

Answer

The solution is \( i(t) = \left(i_0 - \frac{E}{R} \right) e^{-\frac{R}{L} t} + \frac{E}{R} \) and is defined for \( t \in (-\infty, \infty) \).

Step 1: Identify the Type of Differential Equation

The given equation \( L \frac{d i}{d t} + R i = E \) is a first-order linear ordinary differential equation (ODE). It is linear in terms of \( i(t) \) with constant coefficients.

Step 2: Rearrange the ODE

Re-arrange the differential equation to solve it conveniently: \( \frac{d i}{d t} + \frac{R}{L} i = \frac{E}{L} \). This representation is more suitable for applying standard solution techniques for linear first-order ODEs.

Step 3: Solve the Homogeneous Equation

Consider the homogeneous part \( \frac{d i}{d t} + \frac{R}{L} i = 0 \). Separating variables gives \( \frac{d i}{i} = -\frac{R}{L} dt \). Integrating both sides produces \( \ln |i| = - \frac{R}{L} t + C \). Exponentiating both sides yields \( i_h(t) = C e^{-\frac{R}{L} t} \).

Step 4: Find a Particular Solution

To find a particular solution, assume a constant solution \( i_p(t) = i_p \) since the non-homogeneous term \( \frac{E}{L} \) is a constant. Substitute into the equation: \( 0 + \frac{R}{L} i_p = \frac{E}{L} \), giving \( i_p = \frac{E}{R} \).

Step 5: Combine Solutions

The general solution is the sum of the homogeneous solution and the particular solution: \( i(t) = C e^{-\frac{R}{L} t} + \frac{E}{R} \).

Step 6: Apply the Initial Condition

Use the initial condition \( i(0) = i_0 \) to find the constant \( C \). Substitute \( t = 0 \) and \( i(0) = i_0 \) into the general solution: \( i_0 = C + \frac{E}{R} \). Solving for \( C \) gives \( C = i_0 - \frac{E}{R} \).

Step 7: Write the Final Solution

Substitute \( C \) back into the general solution to get \( i(t) = \left(i_0 - \frac{E}{R} \right) e^{-\frac{R}{L} t} + \frac{E}{R} \).

Step 8: Determine the Interval of the Solution

For a linear first-order ODE with constant coefficients, the solution is defined for all \( t \). Thus, the largest interval \( I \) over which the solution is defined is \( (-\infty, \infty) \).

Key Concepts

Initial-Value Problems

When you're tackling an initial-value problem, you're dealing with a differential equation that has been provided with a specific point or condition, called the initial condition. In the exercise example, we have the initial condition \( i(0) = i_0 \). This condition is crucial because it allows us to determine an exact solution out of many possible solutions to the differential equation.

• Initial-value problems help specify which solution we need by giving us the value of the function at a specific point.
• These problems are common in physics and engineering where initial states are known.

The initial condition effectively helps to pin down where the solution starts, which in turn makes it possible to solve for constants involved in the solution. Without this, our solution would still contain arbitrary constants, and we wouldn't have a specific function describing our solution.
In steps where we address this condition, like calculating the constant \( C \) in the solution, we ensure that the final expression accurately reflects the specific case of the system being analyzed.

Linear Differential Equations

Linear differential equations are a specific type of ordinary differential equation that are crucial in mathematical modeling. In the problem presented, the differential equation \( L \frac{d i}{d t} + R i = E \) qualifies as a first-order linear differential equation.

• Linear differential equations have solutions that don’t involve products or powers of the unknown function or its derivatives.
• They often feature constant coefficients, which makes them easier to solve using methods like integrating factors or characteristic equations.

These equations are widely used because they appear in many natural phenomena and engineering problems. For this specific problem, recognizing the linearity allows us to apply systematic methods to find solutions, splitting the problem into finding homogeneous solutions and particular solutions.
The linearity also guarantees that the resulting solutions can be easily analyzed, as the superposition principle applies—meaning you can add solutions together to get a general solution.

Homogeneous Solutions

Solving for the homogeneous solution is one of the key steps when tackling linear differential equations. In the case of the differential equation \( \frac{d i}{d t} + \frac{R}{L} i = 0 \), we are focusing on the part of the system where no external forces or impulses are acting (like when \( E = 0 \)).

• Homogeneous solutions reflect the system's natural behavior without external influences.
• Finding these solutions often involves methods like separation of variables.

In our example, we separated variables and integrated both sides, leading to the natural exponential decay solution \( i_h(t) = C e^{-\frac{R}{L} t} \).
This step is foundational because it reveals the intrinsic properties of the system, governed by things like resistive forces, inertia, etc. Later, we combine these solutions with particular solutions to get a full picture of how the system behaves in the presence of external inputs, yielding the general solution. Knowing the homogeneous part helps you understand the behavior over time, especially how the system might return to equilibrium.