In tuning a stereo system to a radio station, we adjust the tuning control (turn a knob) that changes C (or perhaps L) in an RLC-circuit so that the amplitude of the steady-state current becomes maximum. For what C will this happen?

Reduce to first order and solve, showing each step in detail. xy''+2y'+xy=0, y_1=(cos x)/x

If $f, g$ are linearly dependent on an interval $I$, show that this also holds for any subinterval $J$ of $I$. Is the same true for linear independence? (Give reasons and examples.)

Find a (real) general solution. Mention the rule are you using? (Show each step of your calculation.)

$y^{\prime \prime}+4 y^{\prime}+4 y=e^{-2 x} \sin 2 x$

A 30.0-mH inductor has a reactance of $2.10 \mathrm { k } \Omega$

What is the frequency of the ac current that passes through the inductor?

Find a general solution. (Show the details of your calculations.)

$$\left(D^2+D+1\right) y=\cos x+13 \cos 2 x$$

Factor as in the text and solve. (D²+4.00D+3.36l)y=0

Find the current when $L=0.5 \mathrm{H}$, $C=8 \cdot 10^{-4} \mathrm{~F}, E=t^2 \mathrm{~V}$ and initial current and charge zero.

Why is it nonsense to talk about linear dependence or independence of two functions at a point?

Verify by substitution that the given function form a basis. Solve the given initial value problem. (Show the details of your work.)

$x^2 y^{\prime \prime}-7 y^{\prime}+15 y=0, x^3, x^5, y(1)=0.4$. $y^{\prime}(1)=1.0$

Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.

$$(D+6D+9l)y=16e^{-3x}/(x^2+1)$$

Find the Wronskian. Show linear independence by using quotients and confirm it.

$$e^{4.0x}\text{, }e^{-1.5x}$$

Find the Wronskian. Show linear independence by using quotients and confirm it by Theorem 2.

$$x^kcos(lnx),\text{ }x^ksin(lnx)$$

Find a general solution. Check your answer by substitution. 10y''-32y'+25.6y=0