Using software, obtain a symbolic solution of \(i R+L \frac{\mathrm{d} i}{\mathrm{~d} t}=E\) where \(R, L\) and \(E\) are constants subject to the initial condition \(i(0)=0\).

Short Answer

Expert verified
Question: Solve the first-order linear ordinary differential equation $iR + L\frac{di}{dt} = E$ with the initial condition $i(0) = 0$. Answer: The solution is given by $i(t) = \frac{E}{R}\left(1 - e^{-\frac{Rt}{L}}\right)$.

Step by step solution

01

Identify the First-Order Linear ODE

The given equation is already an ODE. It is in the form of \(L \frac{di}{dt} + Ri = E\), where \(i(t)\) is the current as a function of time, \(t\). Note that \(R, L,\) and \(E\) are constants.
02

Perform Variable Separation

To perform variable separation, we'll first rewrite the ODE as: \(\frac{di}{dt} + \frac{R}{L}i = \frac{E}{L}\) Next, we will use an integrating factor. The integrating factor, denoted by \(IF\), is given by: \(IF = e^{\int \frac{R}{L} dt} = e^{\frac{Rt}{L}}\) Multiply both sides of the ODE by the integrating factor: \(e^{\frac{Rt}{L}}\frac{di}{dt} + \frac{R}{L} e^{\frac{Rt}{L}}i = e^{\frac{Rt}{L}}\frac{E}{L}\) After multiplication, the left-hand side of the equation becomes a perfect derivative: \(\frac{d}{dt} \left( i e^{\frac{Rt}{L}} \right) = e^{\frac{Rt}{L}}\frac{E}{L}\)
03

Integrate Both Sides

Integrate both sides of the equation with respect to time: \(\int \frac{d}{dt} \left( i e^{\frac{Rt}{L}} \right) dt = \int e^{\frac{Rt}{L}}\frac{E}{L} dt\) After integrating, we get: \(i e^{\frac{Rt}{L}} = \frac{E}{R}\left(e^{\frac{Rt}{L}} - 1\right) + C\)
04

Apply the Initial Condition

To find the constant of integration \(C\), we apply the initial condition: \(i(0)=0\). \(0 = \frac{E}{R}\left(e^{\frac{R(0)}{L}} - 1\right) + C\) Solving for C, we find: \(C = \frac{-E}{R}\)
05

Find the Solution for \(i(t)\)

Plug the value of \(C\) back into the equation: \(i e^{\frac{Rt}{L}} = \frac{E}{R}\left(e^{\frac{Rt}{L}} - 1\right) - \frac{E}{R}\) Now, isolate \(i(t)\): \(i(t) = \frac{E}{R}\left(1 - e^{-\frac{Rt}{L}}\right)\) So, the symbolic solution to the given first-order linear ODE, subject to the initial condition \(i(0)=0\), is: \(i(t) = \frac{E}{R}\left(1 - e^{-\frac{Rt}{L}}\right)\)

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