Consider an electric circuit as in Example 5 (page 318 of Apostol). Assume the voltage at time is given by

where and are positive constants. If , prove that the current at time is given by

where depends only on , and . Prove that when .

*Proof.* By Kirchhoff’s law (page 317 of Apostol) we have

So, if and , where and are positive constants, we have

Then, let

(By the Pythagorean identity we can make this choice.) Therefore,