Math 225 Differential Equations I Assignment 1 PDF

Math 225 - Differential Equations I Assignment # 1 Instructions: You are being evaluated on the presentation, as well as the correctness, of your answers. Try to answer questions in a clear, direct, and efficient way. Sloppy or incorrect use of technical terms will...

Math 225 - Differential Equations I Assignment # 1 Instructions: You are being evaluated on the presentation, as well as the correctness, of your answers. Try to answer questions in a clear, direct, and efficient way. Sloppy or incorrect use of technical terms will lower your mark. 1. Which of the equations below are linear, and which are nonlinear? For the nonlinear ones, circle the terms that make the equation nonlinear. For the linear ones, rearrange the equation so that it fits the format shown in class (equation (7) on p 4 in the text), and circle the coefficients of the dependent variable and its derivatives. d2 x d4 θ dθ (a) + sin(t)x = 0 (b) = − ert θ, dt2 dt 4 dt dy dz (c) 2ty + y = t2 y (d) = cos(β) dt dβ d2 y dy (e) 2 − cos(θ) = 3y 2 dθ dθ 2. Consider a population N (t) that is changing according to the following rules: ˆ the per capita birth rate is a constant, 2 ˆ the per capita death rate is an increasing function of the population, 0.25N ˆ the population is harvested at a constant rate, H (a) Using these rules, write the ODE that describes the rate of change of the population. (b) For what value of N is the rate of change of the population equal to zero? Your answer will be a function of H. (c) Sketch direction fields for the ODE when 0 ≤ H ≤ 4, i.e., for H = 0, H = 2, and H = 4 (draw a separate direction field for each value of H). 3. (Section 1.3 #2) The direction field for dy = 2x + y (1) dx is shown in Figure 1. Answer the following questions about this direction field. (a) Sketch the solution curve that passes through (0, −2). From this sketch, write the equation for the solution. (b) Sketch the solution curve that passes through (−1, 3). (c) What can you say about the solution in part (b) as x → +∞? How about x → −∞? 1  Figure 1: Direction field for question #3. 4. Consider the differential equation dx x(x − 2) =. dt t Use the method of isoclines to draw the direction field in the window 0 < t < 4 and x > 0. 5. (Section 2.2 # 2, # 4, & # 6) Consider each the differential equation below. Is it separable? Why or why not? dy (a) = 4y 2 − 3y + 1 dx dy yex+y (b) = 2 dx x +2 ds s+1 (c)s2 + = dt st 6. (Section 2.2 #23) Solve the initial value problem dy π = 2t cos2 (y), y(0) =. dt 4 7. (Section 2.2 #30) Separation of the equation dy = g(x)p(x) dx requires division by p(y), and this may disguise the fact that the roots of the equation p(y) = 0 are actually constant solutions to the ODE. (a) To explore this further, separate the equation dy = (x − 3)(y + 1)2/3 (2) dx 2  to derive the solution 3 x2  y = −1 + −x+C. (3) 6 (b) Show that y ≡ −1 satisfies the original equation (2). (c) Show that there is no choice of the constant C in (3) that will make the solution (3) yield the solution y ≡ −1. Thus, we lost the solution y ≡ −1 when we divided by (y + 1)2/3. 3