Fall 2008 midterm solutions
- 1. MATH2860U Midterm October, 2008 2 1. (4 marks each; total 12 marks) Answer each question in the space provided. You MUST show all of your work. a) Consider the following linear first-order differential equation: xy′ − 3 y = sin x 3 . What is the appropriate integrating factor to help solve this equation? (Just find the integrating factor, but do NOT actually solve the equation). m b) Find all values of m so that x is a solution of the differential equation x 2 y′′ + 2 xy′ − 2 y = 0 c) Find the general solution of the differential equation ( D + 1)( D 2 + 9)( D 2 − 16) 2 y = 0 .
- 2. MATH2860U Midterm October, 2008 3 2. (7 marks) Find the general solution of the differential equation ⎛ 1 ⎞ dy (4 x 3 y 2 − e − x ) + ⎜ 2 x 4 y + ⎟ = 0 ⎜ ⎝ y ⎟ dx ⎠
- 3. MATH2860U Midterm October, 2008 4 3. (Total 12 marks) Write your final answer on the line provided. You do NOT have to show your work, but it is suggested for parts f and g (for part marks in case the final answer is wrong). For all other parts, ONLY the final answer will be marked. a) (2 marks) Suppose that we have a radioactive series in which element A decays to element B then C with rate constants k1 , k 2 as shown below ( k1 , k 2 positive). dA The differential equation governing the amount of element A is = − k1 A . dt What are the differential equations governing the amount of element B and element C? Differential Equation for Element B: _________________________________ Differential Equation for Element C: _________________________________ b) (1 mark) Determine whether the functions y1 = x 2 and y 2 = x 2 + 1 are linearly dependent or linearly independent. (Answer: “Dependent” or “Independent”) Answer : ________________________________________ c) (1 mark) Which of the following statements describes the differential equation x 3 y ′′ + (cos x) y − x 2 e x = 0 ? i. Linear and homogeneous ii. Linear and nonhomogeneous iii. Nonlinear Answer: _________________________________ d) (3 marks) For the equation y′′ + y′ − 12 y = 6 x 2 + 2e −4 x , if it is known −4 x that yc = c1e + c2 e , what is the appropriate form to be assumed for the 3x particular solution, if solving using the method of undetermined coefficients? You do NOT have to solve for the values of the constants A, B, C, etc. y p is of the form: _________________________________
- 4. MATH2860U Midterm October, 2008 5 e) (1 mark) Without attempting to solve the ODE, determine whether the ODE y′ = x 4 − y 2 is guaranteed to have a unique solution through the point (2,0). Answer “True” or “False”. Answer: _________________________________ f) (2 marks) Suppose an LRC circuit is critically damped. Increasing the resistance of the resistor will result in what type of motion? i) underdamped motion. ii) overdamped motion. iii) pure resonant motion iv) both a and c Answer: _________________________________ g) (2 marks) Consider the differential equation y′ = y x + x subject to the 2 initial condition y(3) = 7. Use Euler’s Method with a step size of h = 0.2 to approximate y(3.2). Answer: _________________________________
- 5. MATH2860U Midterm October, 2008 6 4. (9 marks) A horizontal spring with a mass of 2 kg has damping constant 8, and a force of 6 N is required to keep the spring stretched 0.5 m beyond its natural length. The spring is stretched 1m beyond its natural length (i.e. to the right) and then released with zero velocity. Find the position x(t) of the mass at any time t. Note: There is no external force F(t) applied to the spring.
- 6. MATH2860U Midterm October, 2008 7 5. (10 marks total) Use variation of parameters to obtain the particular solution of the given differential equation y′′ + y′ − 12 y = 8e NOTE: You must use variation of 2x parameters; other methods will NOT receive any marks. Hint: You may use the fact that yc = c1e −4 x + c2e3 x for this differential equation.