If R is the resistance, L is the coefficient of…
If $R$ is the resistance, $L$ is the coefficient of self-inductance, and $U$ is the voltage of an electrical circuit, its dynamic equilibrium equation is $L\dfrac{di}{dt}+Ri=U$. Find the current as a function of time $t$ if $i=i_0$ when $t=0$.
Explanation of Method
Divide by $L$ to get standard linear form $\dfrac{di}{dt}+\dfrac{R}{L}i=\dfrac{U}{L}$ and solve with the integrating factor $e^{Rt/L}$.
Step 1: Apply the integrating factor
With $\mu(t)=e^{Rt/L}$: $\dfrac{d}{dt}\!\left(e^{Rt/L}i\right)=\dfrac{U}{L}e^{Rt/L}$.
Step 2: Integrate and solve for $i$
$e^{Rt/L}i=\dfrac{U}{R}e^{Rt/L}+C$, so $i=\dfrac{U}{R}+Ce^{-Rt/L}$. At $t=0$: $i_0=\dfrac{U}{R}+C\Rightarrow C=i_0-\dfrac{U}{R}$.
Result
Final answer:
\[\boxed{\,i(t)=\frac{U}{R}+\left(i_0-\frac{U}{R}\right)e^{-Rt/L}.}\]
The current approaches the steady-state value $U/R$ with time constant $\tau=L/R$.