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MATH 2Z03: Midterm Test 1 Section 1 Instructor: Dmitry Pelinovsky
Instructions: This test paper contains 10 questions (both multiple choice and short answer) printed on both sides of the page. The questions are on pages 2 through 7. Page 8 is available for rough work. You are responsible for ensuring that your copy of the paper is complete. Bring any discrepancies to the attention of the invigilator.
In case of multiple-choice questions, select the one correct answer to each question. All answers must be completed in permanent ink. Point values are indicated. The test is graded out of 34. The CASIO FX 991 calculator only is allowed on the test.
1. (3 marks) Consider the following differential equation y 00 + x 2 y 0 = sin(x)
Circle those characteristics that categorize this equation: First-order Second-order Linear Nonlinear Homogeneous Non-homogeneous
2. (3 marks) For the autonomous differential equation
y 0 = (1 + y 2 ) cos2 (πy) − sin2 (πy)
which one of the following statements is true?
(a) y = 0 is an unstable equilibrium solution.
(b) y = 0.25 is an unstable equilibrium solution.
(c) y = 0 is a stable equilibrium solution.
(d) y = 0.25 is a stable equilibrium solution.
3. (3 marks) Find the integrating factor for the differential equation: xy0 + 6x 3 y = e x − 1
4. (3 marks) Find the roots of the auxiliary equation of y 00 + 3y 0 − 10y = 0.
5. (3 marks) The temperature outside is −10 degrees. You bring an icicle from outside into a room maintained at a constant temperature of 20 degrees. The temperature of the icicle at time t is given by T(t) and it obeys Newton’s law of Cooling:
T 0 (t) = −k(T − T∗),
where T∗ is the room temperature and k > 0 is a constant. After 1 minute, the icicle is at −5 degrees. At what time (measured from when the icicle is first brought inside) will the icicle reach 0 degrees?
6. (3 marks) Consider the differential equation:
y 00 + ay0 + by = 0,
where a and b are constant coefficients. Find the values of a and b given that y = e mx with m = 4 is the only solution which can be written exclusively in terms of the exponential function.
7. (5 marks) Find two particular (real) solutions to the differential equation: y·· + 4y·+ 9y = 0,
and show that these solutions are linearly independent at x = 0 by computing their Wronskian at x = 0.
8. (5 marks) Consider the following differential equation y·= 2xy2
subject to the initial condition y(0) = 4. Find the unique solution of the initial-value problem and specify for what values of x it is defined.
9. (3 marks) Consider the differential equation y 0 + p(x)y = q(x) and assume that this equation has the following two particular solutions
y1(x) = 2e x + 1 − x, y2(x) = 1 − e x − x.
Write the general solution of this equation and explain why it is a general solution.
10. (3 marks) Consider the differential equation y·+ 2y = x 2 + 1
Find the particular solution of this equation in the polynomial form
y(x) = Ax2 + Bx + C
with coefficients A, B, and C to be determined.