3. Identify and find the general solution of each of the following first order differential equations.

Transcription

1 Final Exam MATH 33, Sample Questions. Fall 7. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C x is the general solution of a differential equation. Find the equation. Answer: x y xy + y = 3. Identify and find the general solution of each of the following first order differential equations. a x y + 3 y = cos x x. Answer: Linear, b y = y + 3 4y + xy. y = sin x x 3 + C x 3 Answer: Separable, y = C4 + x 3 c x dy xy dx = x 4 cos x dx. Answer: Linear, y = x sin x + Cx d yy = xy x y +. Answer: Separable, y = Ce x + e xy y = x ln x. Answer: Linear, y = xlnx + Cx f ln x dy dx = y x. Answer: Separable, y = C ln x 4. Find the solution of the initial-value problem xy + 3y = ex x, y = Answer: y = ex x ex x 3 + x 3 5. If y = yx is the solution of the initial-value problem y + 3y = 3e x, y =, then lim yx = x Answer: /3

2 6. Find the solution of the initial-value problem y = x y y y +, y3 = Answer: y + lny = 3 x3 x 5 7. Given the one-parameter family y 3 = Cx + 4. a Find the differential equation for the family. b Find the differential equation for the family of orthogonal trajectories. c Find the family of orthogonal trajectories. Answer: a y = y3 8 3xy. 8. Given the one-parameter family y = Ce x +. a Find the differential equation for the family. b y = 3xy y 3 8. c 3x y + y 3 + Cy + 6 =. b Find the differential equation for the family of orthogonal trajectories. c Find the family of orthogonal trajectories. Answer: a y = y. b y = y. c x + y = C. 9. A certain radioactive material is decaying at a rate proportional to the amount present. If a sample of grams of the material was present initially and after 3 hours the sample lost 3% of its mass, find: a An expression for the mass At of the material remaining at any time t. b The mass of the material after 8 hours. c The half-life of the material. t/3 8/ ln Answer: a At = b A8 = c t = ln 7/. Scientists observed that a colony of penguins on a remote Antarctic island obeys the population growth law. There were penguins in the initial population and there were 3 penguins 4 years later. a Give an expression for the number Pt of penguins at any time t. b How many penguins will there be after 6 years? c How long will it take for the number of penguins to quadruple? Answer: a Pt = e t 4 ln3 = 3 t/4 b P6 = 3 3/. c t = 4 ln4 ln3 years.

3 . Determine a fundamental set of solutions of y y 5 y =. Answer: { e 5x, e 3x}. Find the general solution of y + 6y + 9y =. Answer: y = C e 3x + C xe 3x. 3. Find the solution of the initial-value problem y 7y + y =, y = 3, y =. Answer: y = e 3x 9e 4x. 4. The function y = e 3x sin x is a solution of a second order, linear, homogeneous differential equation with constant coefficients. What is the equation? Answer: y + 6y + 3y = 5. The function y = e 3x + 4 xe 3x is a solution of a second order, linear, homogeneous differential equation with constant coefficients. What is the equation? Answer: y 6y + 9y = 6. Find a particular solution of y 6y + 8y = 4e 4x. Answer: z = xe 4x. 7. Give the form of a particular solution of the nonhomogeneous differential equation y 8 y + 6 y = e 4x + 3 cos 4x x +. Answer: z = Ax e 4x + B cos 4x + C sin 4x + Dx + E. 8. Given the differential equation y 4 y + 4 y = ex x a Give the general solution of the reduced equation. b Find a particular solution of the nonhomogeneous equation. Answer: a y = C e x + C xe x. b z = xe x + x e x ln x or z = x e x ln x. 9. Find the general solution of y 4 x y + 6 x y = 4. HINT: The reduced equation has solutions x of the form y = x r. Answer: y = C x + C x Find a particular solution of y + 4y = tan x. Answer: y = C cos x + C sin x cos x ln[sec x + tan x] 3

4 . The general solution of y 4 6 y + 7 y 8 y + y = is: HINT: is a root of the characteristic polynomial Answer: y = C e x + C xe x + C 3 e x cos x + C 4 e x sin x.. Give the form of a particular solution of y y 36 y = x + 3e x sin 3x Answer: z = Ax + B + Cxe x + Dx cos 3x + Ex sin 3x. 3. Find the Laplace transform of the solution of the initial-value problem y y 6 y = 3; y = 5, y = Answer: Y = 3 ss s 6 + 5s 5 s s Find fx = L [Fs] if Fs = 3 s + 4s + 3 s + 4. Answer: fx = 3x + 4 cos x + 3 sin x 5. Find L[fx] if fx = { x + x x < 4 x x 4 Answer: Fs = s 3 + s e 4s s 3 9e 4s s e 4s s. 6. Fs = 5 s 3 s e 3s s + 3 e 3s s + s + e 3s s + π. Find L [Fs] = fx. 5x, x < 3 Answer: fx = 3x + 6 cos πx π sin πx, x Given the initial-value problem y 4 y = e x, y = 3. a Find the Laplace transform of the solution. b Find the solution by finding the inverse Laplace transform of your answer to a. Answer: a Y = s + s b y = s 4 3 e4x 3 e x. 4

5 8. Given the system of equations x +y z = x +5y 4z = 3 x y z = a Write the augmented matrix for the system. b Reduce the augmented matrix to row-echelon form. c Give the solution set of the system. Answer: a b c x = 3a, x = + a, x 3 = a, a arbitrary.. 9. For what values of k, does the system of equations x + y + 3z = 4 y + 5z = 9 x + 3y + k 8z = k + have a a unique solution? b infinitely many solutions? c no solutions? Answer: a k ±3 b k = 3 c k = 3 3. Given the system of equations x y = x y +kz = 3 y z = k The values of k, if any, such that the system has infinitely many solutions is are: Answer: No values of k. 3. Find the values of λ, if any such that A = Answer: λ, 3 3. The matrix A = Answer: A = λ 3 λ 5 is nonsingular. Find A.. is nonsingular. 5

6 33. The system of equations x y + 3z = 4 y + z = x + z = has a unique solution. Find y. Answer: y = 34. Determine whether the vectors v =, 3,, v =,,, v 3 =, 5,, v 4 =, 4, 4 are linearly dependent or linearly independent. If they are linearly dependent, find the maximal number of independent vectors. Answer: Linearly dependent; the maximum number of independent vectors is Find the eigenvalues and eigenvectors of 3. Hint: 6 is an eigenvalue. Answer: λ = 6, ; λ =, 36. Find the eigenvalues and eigenvectors of Answer: λ =, ; λ = λ 3 =, ; λ 3 =, Find the solution of the initial-value problem x = is an eigenvalue. Answer: xt = e t e t 38. Find a fundamental set of solutions of x = Answer: {e 3t [ cos t sin t. ] x., e 3t [cos t 3. Hint: is an eigenvalue. x, x = + sin t. HINT: ]}. 6

7 39. Find the general solution of x = Answer: xt = C e 3t 4. Find the general solution of x = polynomial. Answer: xt = C e t C [e 3t x. + C e t 4. Find a fundamental set of solutions of x = characteristic polynomial. Answer: et, e t, e t + te 3t ] x. HINT: is a root of the characteristic + C e t + te t. x. HINT: is a root of the 7