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### 219hw4

Course: MAE EE IEN mae ee ien, Spring 2009
School: WVU
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Word Count: 184

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219, Math Homework 4 Due date: 30.12.2005, Wednesday Suppose that K &gt; 0, and f (t) is defined as 1 0 if 4n t &lt; 4n + 1 otherwise f (t) = where n runs through the set of integers. (a) Determine the Fourier series for f (t). (b) Consider the differential equation d2 x + x = f (t). dt2 By using ODE Architect, solve (and graph the solutions of) this equation for as many values of K as possible between...

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219, Math Homework 4 Due date: 30.12.2005, Wednesday Suppose that K > 0, and f (t) is defined as 1 0 if 4n t < 4n + 1 otherwise f (t) = where n runs through the set of integers. (a) Determine the Fourier series for f (t). (b) Consider the differential equation d2 x + x = f (t). dt2 By using ODE Architect, solve (and graph the solutions of) this equation for as many values of K as possible between 0.5 and 10 for 0 t 100.(You can enter f (t) in ODE Architect using the built in command SqW ave(t, K) L, and then set L = 4 K, and assign K definite values on the lines below). Record the maximum values of x(t) for each of these K's, and plot a K vs. max(x(t)) graph by hand. (c) Which values of K result in a resonance in the system? (Hint: you should find 5 such values). Can you relate these values to the Fourier series terms? Please print the resonance graphs. (d) What do the Fourier coefficients correspond to on the resonance graphs? We wish you a happy new year and success on your finals!
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WVU - MAE EE IEN - mae ee ien
- CHAPTER 1. -Chapter OneSection 1.1 1.For C &quot;&amp; , the slopes are negative, and hence the solutions decrease. For C &quot;&amp; , the slopes are positive, and hence the solutions increase. The equilibrium solution appears to be Ca&gt;b oe &quot;&amp; , to which all
WVU - MAE EE IEN - mae ee ien
- CHAPTER 2. -Chapter TwoSection 2.1 1a+ba,b Based on the direction field, all solutions seem to converge to a specific increasing function. a- b The integrating factor is .a&gt;b oe /\$&gt; , and hence Ca&gt;b oe &gt;\$ &quot;* /#&gt; - /\$&gt; It follows that all s
WVU - MAE EE IEN - mae ee ien
- CHAPTER 3. -Chapter ThreeSection 3.1 1. Let C oe /&lt;&gt; , so that C w oe &lt; /&lt;&gt; and C ww oe &lt; /&lt;&gt; . Direct substitution into the differential equation yields a&lt;# #&lt; \$b/&lt;&gt; oe ! . Canceling the exponential, the characteristic equation is &lt;# #&lt; \$ o
WVU - MAE EE IEN - mae ee ien
- CHAPTER 4. -Chapter FourSection 4.1 1. The differential equation is in standard form. Its coefficients, as well as the function 1a&gt;b oe &gt; , are continuous everywhere. Hence solutions are valid on the entire real line. 3. Writing the equation in
WVU - MAE EE IEN - mae ee ien
- CHAPTER 5. -Chapter FiveSection 5.1 1. Apply the ratio test : lim aB \$b8&quot; k a B \$b 8 kHence the series converges absolutely for kB \$k &quot; . The radius of convergence is 3 oe &quot; . The series diverges for B oe # and B oe % , since the n-th ter
WVU - MAE EE IEN - mae ee ien
- CHAPTER 6. -Chapter SixSection 6.1 3.The function 0 a&gt;b is continuous. 4.The function 0 a&gt;b has a jump discontinuity at &gt; oe &quot; . 7. Integration is a linear operation. It follows that (E !-9=2 ,&gt; /=&gt; .&gt; oe&quot; E ,&gt; =&gt; &quot; E ,&gt; =&gt; ( / / .&gt;
WVU - MAE EE IEN - mae ee ien
- CHAPTER 7. -Chapter SevenSection 7.1 1. Introduce the variables B&quot; oe ? and B# oe ? w . It follows that B&quot;w oe B# and B#w oe ? ww oe #? !&amp; ? w . In terms of the new variables, we obtain the system of two first order ODEs B&quot;w oe B# B#w oe #B&quot;
WVU - MAE EE IEN - mae ee ien
- CHAPTER 8. -Chapter EightSection 8.1 2. The Euler formula for this problem is C8&quot; oe C8 2^&amp; &gt;8 \$C8 , C8&quot; oe C8 &amp;82# \$2 C8 ,in which &gt;8 oe &gt;! 82 Since &gt;! oe ! , we can also writea+b. Euler method with 2 oe !&amp; &gt;8 C8 8oe# !&quot; &quot;&amp;*)! 8oe% !
WVU - MAE EE IEN - mae ee ien
- CHAPTER 9. -Chapter NineSection 9.1 2a+b Setting x oe 0 /&lt;&gt; results in the algebraic equations OE &amp;&lt; \$For a nonzero solution, we must have ./&gt;aA &lt; Ib oe &lt;# ' &lt; ) oe ! . The roots of the characteristic equation are &lt;&quot; oe # and &lt;# oe % . For
WVU - MAE EE IEN - mae ee ien
- CHAPTER 10. -Chapter TenSection 10.1 1. The general solution of the ODE is CaBb oe -&quot; -9= B -# =38 B Imposing the first boundary condition, it is necessary that -&quot; oe ! . Therefore CaBb oe -# =38 B . Taking its derivative, C w aBb oe -# -9= B
WVU - MAE EE IEN - mae ee ien
- CHAPTER 11. -Chapter ElevenSection 11.1 1. Since the right hand sides of the ODE and the boundary conditions are all zero, the boundary value problem is homogeneous. 3. The right hand side of the ODE is nonzero. Therefore the boundary value prob
WVU - MAE EE IEN - mae ee ien
CHAPTER 3 ENGINEERING CIRCUIT ANALYSIS 1. 3. 5. 7. 9. 11. Circuit diagram not shown. (a) 4 nodes; (b) 5 branches; (c) yes, path; no, loop. (a) 4; (b) 5; (c) yes,no,yes,no,no (a) 3 A; (b) -3 A; (c) 0 ix = 1 A; iy = 5 A.SELECTED ANSWERSIf the DMM a
WVU - MAE EE IEN - mae ee ien
Solution to Problem Set IChapter 2, Problem 4. A certain 15 V dry-cell battery, completely discharged, requires a current of 100 mA for 3 hr to completely recharge. What is the energy storage capacity of the battery, assuming the voltage does not de
WVU - MAE EE IEN - mae ee ien
Solution to Problem Set IIChapter 2, Problem 35. The circuit of Fig. 2.38 is constructed so that vS = 2 sin 5t V, and r = 80 . Calculate vout at t = 0 and t = 314 ms.+ vS+ 1+ 103v 1k v out rv FIGURE 2.38Chapter 2, Solution 35. vout =
WVU - MAE EE IEN - mae ee ien
Solution to Problem Set IIIChapter 3, Problem 35. Find the power absorbed by each circuit element of Fig. 3.68 if the control for the dependent source is (a) 0.8ix ; (b) 0.8iy . In each case, demonstrate that the absorbed power quantities sum to zer
WVU - MAE EE IEN - mae ee ien
Solution to Problem Set IVChapter 4, Problem 2. (a) Find vA, vB, and vC if vA + vB + vC = 27, 2vB + 16 = vA - 3vC, and 4vC + 2vA + 6 = 0. (b) Evaluate the determinant:B B B10 1 2 3 1 2 3 4 2 3 4 1 3 4 1 2Chapter 4, Solution 2. (a) 1 -1 2 1 2
WVU - MAE EE IEN - mae ee ien
Solution to Problem Set IXChapter 10, Problem 2. (a) If -10 cos t + 4 sin t = A cos( t + ) , where A &gt; 0 and - 180 o &lt; 180 o , find A and . (b) If 200 cos(5t + 130 o ) = F cos 5t + G sin 5t , find F and G. (c) Find three values of t, 0 t 1 s
WVU - MAE EE IEN - mae ee ien
Solution to Problem Set VChapter 4, Problem 36. Determine each mesh current in the circuit of Fig. 4.65.12 0.1vx 5 + 6V+ 11 vx +12 2V 3 +1.5 VFIGURE 4.65Chapter 4, Solution 36. We define a clockwise mesh current i3 in the upper righ
WVU - MAE EE IEN - mae ee ien
Solution to Problem Set VIChapter 5, Problem 20. With the assistance of the method of source transformations, (a) convert the circuit of Fig. 5.65a to a single independent voltage source in series with an appropriately sized resistor; and (b) conver
WVU - MAE EE IEN - mae ee ien
Solution to Problem Set VIIChapter 5, Problem 47. (a) Find the Thvenin equivalent of the network shown in Fig. 5.89. (b) What power would be delivered to a load of 100 at a and b?40 100 a +120 V200 i11.5i1bFIGURE 5.89Chapter 5, Solu
WVU - MAE EE IEN - mae ee ien
Solution to Problem Set VIIIChapter 7, Problem 3. Calculate the current flowing through a 1 mF capacitor in response to a voltage v across its terminals if v equals: (a) 30te t V; (b) 4e 5t sin 100t V.1Chapter 7, Solution 3.i=C dv dt(a) (b)
WVU - MAE EE IEN - mae ee ien
Solution to Problem Set XChapter 10, Problem 41. Find Zin at terminals a and b in Fig. 10.58 if equals (a) 800 rad/s; (b) 1600 rad/s.2 F a 300 bFIGURE 10.5816000.6 HChapter 10, Solution 41. (a) = 800 : 2F - j 625, 0.6H j 480 300(- j 6
WVU - MAE EE IEN - mae ee ien
Solution to Problem Set XIChapter 11, Problem 12.1Calculate the average power generated by each source and the average power delivered to each impedance in the circuit of Fig. 11.30.10 A5 508 20j10 AFIGURE 11.30Chapter 11, Solution
WVU - MAE EE IEN - mae ee ien
1. A tank is filled with seawater to a depth of 12 ft. If the specific gravity of seawater is 1.03 and the atmospheric pressure at this location is 14.8 psi, the absolute pressure (psi) at the bottom of the tank is most nearly A. 5.4 B. 20.2 C. 26.8
WVU - MAE EE IEN - mae ee ien
1. A tank is filled with seawater to a depth of 12 ft. If the specific gravity of seawater is 1.03 and the atmospheric pressure at this location is 14.8 psi, the absolute pressure (psi) at the bottom of the tank is most nearly A. 5.4 B. 20.2 C. 26.8
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
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WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien
WVU - MAE EE IEN - mae ee ien