Chapter 20: Problem 2
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 \(i(0)=0\).
Chapter 20: Problem 2
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 \(i(0)=0\).
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Get started for freeThe charge, \(q\), on a capacitor in an \(L C R\) series circuit satisfies the second-order differential equation $$ L \frac{\mathrm{d}^{2} q}{\mathrm{~d} t^{2}}+R \frac{\mathrm{d} q}{\mathrm{~d} t}+\frac{1}{C} q=E $$ where \(L, R, C\) and \(E\) are constants. Show that if \(2 L=C R^{2}\) the general solution of this equation is $$ \begin{aligned} &q= \\ &\mathrm{e}^{-t /(C R)}\left(A \cos \frac{1}{C R} t+B \sin \frac{1}{C R} t\right)+C E \end{aligned} $$ If \(i=\frac{\mathrm{d} q}{\mathrm{~d} t}=0\) and \(q=0\) when \(t=0\) show that the current in the circuit is $$ i=\frac{2 E}{R} \mathrm{e}^{-t /(C R)} \sin \frac{1}{C R} t $$
Use Euler's method to find a numerical solution of \(\frac{\mathrm{dy}}{\mathrm{d} x}=\frac{x y}{x^{2}+2}\) subject to \(y(1)=3\). Take \(h=0.1\) and hence approximate \(y(1.5)\). Obtain the true solution using the method of separation of variables. Work throughout to six decimal places.
Find a particular integral for the equation $$ \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+\frac{\mathrm{d} y}{\mathrm{~d} x}+y=1+x $$
Use an integrating factor to obtain the general solution of \(i R+L \frac{\mathrm{d} i}{\mathrm{~d} t}=\sin \omega t\) where \(R, L\) and \(\omega\) are constants.
Show that \(x(t)=7 \cos 3 t-2 \sin 2 t\) is a solution of $$ \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+2 x=-49 \cos 3 t+4 \sin 2 t $$
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