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

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1 Final Exam MATH 33, Sample Questions. Fall 6. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C is the general solution of a differential equation. Find the equation. Answer: xy y = 3. Identify and find the general solution of each of the following first order differential equations. a xy = 5x 3 y / 4y Answer: Bernoulli, y / = x3 + C x b x y + 3 y = cos x x. Answer: Linear, c x dy dx = x + xy + y y = sin x x 3 + C x 3 Answer: Homogeneous, y = x tanln x + C d y = y + 3 4y + xy. Answer: Separable, y = C4 + x 3 e x y = 4x 3 y 3 + xy Answer: Bernoulli, y = x C x 4 f x 3 y = x y + x 3 e y/x Answer: Homogeneous, y = x ln C ln x 4. 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. b y = 3xy y 3 8. c 3x y + y 3 + Cy + 6 =. 5. 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:

2 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/ 6. 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. 7. A disease is infecting a herd of cows. Let Pt be the number of sick cows t days after the outbreak. Suppose that 5 cows had the disease initially, and suppose that the disease is spreading at a rate proportional to the product of the time elapsed and the number of cows who do not have the disease. a Give the mathematical model initial-value problem for P. b Find the general solution of the differential equation in a. c Find the particular solution that satisfies the initial condition. Answer: a dp dt = kt P, P = 5 b Pt = Ce kt / c Pt = 95e kt / 8. Determine a fundamental set of solutions of y y 5 y =. Answer: { e 5x, e 3x} 9. Find the general solution of y + 6y + 9y =. Answer: y = C e 3x + C xe 3x.. Find the solution of the initial-value problem y 7y + y =, y = 3, y =. Answer: y = e 3x 9e 4x.. 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 =

3 . 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 = 3. Find a particular solution of y 6y + 8y = 4e 4x. Answer: z = xe 4x. 4. 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. 5. 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. 6. 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] 8. 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. 9. 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

4 . 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 6.. Find fx = L [Fs] if Fs = 3 s + 4s + 3 s + 4. Answer: fx = 3x + 4 cos x + 3 sin x. 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. 3. 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 = 5. Given the system of equations s + s b y = s 4 3 e4x 3 e x. 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.. 4

5 6. 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. 7. Find the values of λ, if any such that A = Answer: λ, 3 8. The matrix A = Answer: A = 9. The system of equations has a unique solution. Find y. Answer: y = λ 3 λ 5 is nonsingular. Find A.. x y + 3z = 4 y + z = x + z = is nonsingular. 3. 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. 3. Find the eigenvalues and eigenvectors of Answer: λ = 6, ; λ =, 6 3 ; λ 3 =,. Hint: 6 is an eigenvalue. 3 5

6 3. Find the eigenvalues and eigenvectors of Answer: λ =, ; λ = λ 3 =, Find the solution of the initial-value problem x = is an eigenvalue. Answer: xt = e t e t 34. Find a fundamental set of solutions of x = Answer: {e 3t [ cos t 35. Find the general solution of x = Answer: xt = C e 3t 36. Find the general solution of x = polynomial. Answer: xt = C e t 3 4 sin t. 4 + C [e 3t ] 5 4 x. + C e t 37. Find a fundamental set of solutions of x = characteristic polynomial. Answer: et, e t, e t 3 x., e 3t [cos t + te 3t. Hint: is an eigenvalue. x, x = ] + sin t. HINT: ]} x. HINT: is a root of the characteristic + C e t + te t.. x. HINT: is a root of the 6

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

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

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