Calculus III

285 Exam 1 Study Guide 1 Integrating Factor y 0 + P ( t ) y = Q ( t ) μ = e R P ( t ) μy 0 + μP ( t ) y = μQ ( t ) ( μy ) 0 = μQ ( t ) 2 Substitutions y 0 = f  y t  : v ( t )= y t 2 nd Order Diﬀerential Equation: v ( y )= y 0 dv dy dy dt = v dv dy = y 00 or y = e rt Double Root ⇒ y = C 1 e rt + C 2 te rt 2 Diﬀerent Roots ⇒ y = C 1 e r 1 t + C 2 e r 2 t Bernouilli: y 0 + P ( t ) y = Q ( t ) y α v ( t )= y 1 - α 1 3 Phase Diagrams Graph y 0 as a function of y . On the y -axis (“Phase Line”): If the function is positive: Draw: → If the function is negative: Draw: ← How to Analyze Phase Diagram: Look at critical points (zeros of phase diagram): ←→⇒ Unstable →←⇒ Stable ←← or →→⇒ Semi-stable 4 Intervals y ( n ) + P 1 ( t ) y ( n - 1) + ··· + P n - 2 ( t ) y 0 + P n - 1 ( t ) y = f ( t ) Look for discontinuity points from P 1 ( t ) , ... ,P n - 2 ( t ) ,P n - 1 ( t ) ,f ( t ) Construct interval with initial condition, t 0 , inside it. Make sure the interval spans out as far as possible, stopping at discontinuity points. 5 Miscellaneous Homogeneous: All terms have y,y 0 ,y 00 ,... Inhomogeneous: Not all terms have y,y 0 ,y 00 ,... Linear: y ( n ) + P 1 ( t ) y ( n - 1) + ··· + P n - 2 ( t ) y 0 + P n - 1 ( t ) y = f ( t ) Non-linear: Opposite of Linear. 2

MATH | 285 | Calculus III | study guides | 285 Exam 1 Study Guide 1 Integ

Math 285 Final - Practice Full Name: Net ID: UIN: Circle your class section: • You must not communicate with other students during this test. • No written materials of any kind allowed. • No phones, calculators, iPods or electronic devices of any kind allowed for ANY reason, including checking the time (you may use a simple wristwatch). • Do not turn this page until instructed to do so. • There are many di ↵ erent versions of this exam. • Violations of academic integrity (in other words, cheating) will be taken extremely seriously. Fill in the following answers on the scantron form now: 93. A 94. A 95. D 96. C Multiple Choice Questions. Mark answers to questions 1 to 12 on your scantron form 1. (2 points) What is the order of the following di ↵ erential equation for y ( x )? y 4 + sin ( y ) y 0 +( y 00 ) 2 + x 5 =0 (A) Fourth order (B) None of these (C) Second order (D) First order (E) Third order 2. (5 points) The solution of the initial value problem 2 yy 0 =3 e 3 x ,y (0) = - 4 is (A) None of these choices is correct. (B) y = - p e 3 x + 15 (C) y = p e 3 x - 17 (D) y = p e 3 x + 15 (E) y = ± p e 3 x + 15 3. (3 points) Consider the solution curves in the following ﬁgure: 0.5 1.0 1.5 2.0 2.5 3.0 - 2 2 4 6 8 10 Which model equation corresponds to the above ﬁgure? (A) dp dt =3 p (2 - p )(8 - p ) (B) dp dt = - 3 p (2 - p )(8 - p ) (C) dp dt = - 3 p (2 - p )(9 - p ) (D) dp dt =3 p (1 - p )(5 - p )(9 - p ) (E) dp dt =3 p 2 (8 - p ) 4. (5 points) Consider the di ↵ erential equations I: y 00 + xy 0 + x 2 y =4 e x II: xyy 0 y 00 =1 III: yy 0 y 00 =3 IV: yy 00 =3 y 0 V: y 00 + x 2 y 0 = sin x VI: yy 0 + sin( y 00 ) = cos x Which of these, if any, is reducible to a pair of ﬁrst order di ↵ erential equations by a substitution? (A) Each of I and II (B) Each of III, IV, and V (C) Just V (D) None of them (E) Each of IV and VI 5. (2 points) Consider the following di ↵ erential equation for y ( x ): x 2 y 00 + y 0 = x 4 Which of these statements is true? (A) The equation is linear and homogeneous (B) None of these (C) The equation is linear and non-homogeneous (D) The equation is non-linear 6. (5 points) Consider the following oscillator equation for x ( t ): x 00 +4 x = 2 cos ( ! t ) First determine the value of ! so that the oscillator is in resonance and then determine which of the following is the correct form for the particular solution: (A) None of these (B) 1 3 cos (3 t ) (C) 1 2 t sin (2 t ) (D) 1 4 sin (4 t ) (E) 1 4 t sin (4 t ) 7. (5 points) Which choice below is the correct form of a particular solution of the di ↵ erential equation y 000 - y 00 +4 y 0 - 4 y =3 e t + 5 sin 4 t ? (A) Y = Ate t + B cos 4 t + C sin 4 t (B) Y = Ae t + B cos 4 t + C sin 4 t (C) Y = Ae t + Bt cos 4 t + Ct sin 4 t (D) None of these is correct. (E) Y = Ate t + Bt cos 4 t + Ct sin 4 t 8. (3 points) The amplitude/phase shift form of the function f ( x ) = cos ⇡ x - sin ⇡ x is (A) p 2 cos( ⇡ x - ⇡ 4 ) (B) 2 cos( ⇡ x + ⇡ 3 ) (C) 2 cos( ⇡ x + ⇡ 2 ) (D) p 2 cos( ⇡ x + ⇡ 4 ) (E) 2 cos( ⇡ x + ⇡ 6 )

MATH | 285 | Calculus III | study guides | Math 285 Final - Practice Full

Midterm_2_Practice_Solutions.pdf-Math 285 Midterm 2 Practice Full

285 Exam 3 Study Guide 1 Boundary Value Problems Can get no solution, 1 exact solution, or inﬁnite solutions. Solve the problem like usual. If you ﬁnd a contradiction, = ⇒ no solution. If your solution has a constant ( C 1 , etc.), = ⇒ inﬁnite solutions. If you’re at this point, then you should have 1 exact solution (with no constants). 2 Eigenvalues 3 Cases to Test: • λ>γ • λ = γ • λ<γ γ = Critical value that separates diﬀerent types of solutions. Substitutions: • λ>γ λ = γ + α 2 • λ = γ • λ<γ λ = γ - α 2 α ∈ R , α 6 =0 From your 3 cases, try to get a non-trivial solution. If you ﬁnd non-trivial solutions, denote it like so: λ n = ... y n = ... (Remove any constants) 1 3 Fourier Series Given that your function is periodic, with Period = 2 L : f ( t )= A 0 2 + ∞ X n =1 A N cos  nπt L  + B N sin  nπt L  A 0 = 1 L Z L - L f ( t ) dt A N = 1 L Z L - L f ( t ) cos  nπt L  dt B N = 1 L Z L - L f ( t ) sin  nπt L  dt ? The Integrals don’t have to have bounds - L to L , they can have diﬀerent bounds (as long as it covers 1 period of the function). Conceptual Parts: • Cosine Fourier Series: Even Extension of f ( t ). • Sine Fourier Series: Odd Extension of f ( t ). 4 Heat Equation μ t = kμ xx μ ( x, 0) = f ( x ) [Initial Condition] There are 2 cases, which lie under the General Case: 1. Zero Temperature Boundaries 2. Perfectly Insulated Zero Temperature Boundaries: Let μ ( x, t )= X ( x ) T ( t ), with the following boundary conditions: μ (0 ,t )=0 [Zero temperature on one side.] μ ( L, t )=0 [Zero temperature on the other side.] So, μ t = X ( x ) T 0 ( t ) μ xx = X 00 ( x ) T ( t ) 2 We then have: μ t = kμ xx = ⇒ X ( x ) T 0 ( t )= kX 00 ( x ) T ( t )= ⇒ T 0 ( t ) kT ( t ) = X 00 ( x ) X ( x ) = - λ Applying the boundary conditions, we yield: μ (0 ,t )= X (0) T ( t )=0 = ⇒ X (0) = 0 μ ( L, t )= X ( L ) T ( t )=0 = ⇒ X ( L )=0 We are now able to get 2 separate ODEs with new boundary conditions: T 0 ( t )+ λkT ( t )=0 X 00 ( x )+ λX ( x )=0 X (0) = 0 X ( L )=0 Both are eigenvalue problems! Solving the 2 nd equation ﬁrst as usual, we yield: λ n = n 2 π 2 L 2 [ λ> 0] X n ( x ) = sin  nπx L  [ n ∈ Z + ] ? There are no negative or zero eigenvalues (that process was skipped for the convenience of the reader). We can then rewrite the 1 st equation like so: T 0 ( t )+ λkT ( t )=0 = ⇒ T 0 ( t )+  n 2 π 2 L 2 ! kT ( t )=0 The solution to this ODE is given by: T n ( t )= e - k  n 2 π 2 L 2 ! t So, putting it all together: μ n ( x, t )= B n X n ( x ) T n ( t )= B n sin  nπx L  e - k  n 2 π 2 L 2 ! t μ ( x, t )= ∞ ∑ n =1 μ n ( x, t )= ∞ ∑ n =1 B n sin  nπx L  e - k  n 2 π 2 L 2 ! t 3 Applying the initial condition, we have: μ ( x, 0) = ∞ ∑ n =1 B n sin  nπx L  = f ( x ) [A Sine Fourier Series!] B n = 2 L Z L 0 f ( x ) sin  nπx L  dx Perfectly Insulated: Let μ ( x, t )= X ( x ) T ( t ), with the following boundary conditions: μ x (0 ,t )=0 [Zero heat ﬂow on one side.] μ x ( L, t )=0 [Zero heat ﬂow on the other side.] So, μ t = X ( x ) T 0 ( t ) μ xx = X 00 ( x ) T ( t ) We then have: μ t = kμ xx = ⇒ X ( x ) T 0 ( t )= kX 00 ( x ) T ( t )= ⇒ T 0 ( t ) kT ( t ) = X 00 ( x ) X ( x ) = - λ Applying the boundary conditions, we yield: μ x (0 ,t )= X 0 (0) T ( t )=0 = ⇒ X 0 (0) = 0 μ x ( L, t )= X 0 ( L ) T ( t )=0 = ⇒ X 0 ( L )=0 We are now able to get 2 separate ODEs with new boundary conditions: T 0 ( t )+ λkT ( t )=0 X 00 ( x )+ λX ( x )=0 X 0 (0) = 0 X 0 ( L )=0 Both are eigenvalue problems! Solving the 2 nd equation ﬁrst as usual, we yield: λ n = n 2 π 2 L 2 [ λ> 0] X n ( x ) = cos  nπx L  [ n ∈ Z + ] λ 0 =0 X 0 ( x )=1 4

MATH | 285 | Calculus III | study guides | 285 Exam 3 Study Guide 1 Bound

Midterm_3_Practice.pdf-Math 285 Midterm 3 - Practice

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