The weight W of an object varies inversely as the square of the distance d from the center of the earth. At sea level (3978 mi from the center of the earth), an astronaut weighs 115 lb. Find her weight when she is 413 mi above the surface of the earth and the spacecraft is not in motion. Her weight is Ib. (Round your answer to the nearest hundredth.)
Which of the following are first order linear differential equations? A. sin(x)dy dx − 3y = 0 B. xdy dx − 4y = x6 ex C. dP dt + 2tP = P + 4t − 2 D. (dy dx)2 + cos(x)y = 5 E. d2y dx2 + sin(x)dy dx = cos(x) F. dy dx = y2 − 3y
In problems below, (a) identify the independent variable and the dependent variable of each equation (use 't' for the independent variable if an independent variable is not given explicitly); (b) give the order of each differential equation (enter '1' for first order, '2' for second order and so on; do not include the quotes); and (c) state whether the equation is linear or nonlinear. If your answer to (c) is nonlinear, make sure that you can explain why this is true.
Determine the order of the given differential equation and state whether the equation is linear or nonlinear. d4u dr4 + du dr + 5u = cos(r + u) (a) The order of this differential equation is (b) The equation is Choose
It can be helpful to classify a differential equation, so that we can predict the techniques that might help us to find a function which solves the equation. Two classifications are the order of the equation -- (what is the highest number of derivatives involved) and whether or not the equation is linear . Linearity is important because the structure of the the family of solutions to a linear equation is fairly simple. Linear equations can usually be solved completely and explicitly. Determine whether or not each equation is linear: ? d2y dt2 + sin(t+y) = sint d3y dt3 + tdy dt + (cos2(t))y = t3 (1 + y2)d2y dt2 + tdy dt + y = et y′′ − y + t2 = 0
Check all that apply to the differential equation dy dx − sin2(x)y = cos2(x)y + x A. ODE B. PDE C. first order D. second order E. third order F. linear G. a candidate for integrating factors H. separable I. homogeneous J. autonomous
Consider the slope field shown. (a) For the solution that satisfies y(0) = 0, sketch the solution curve and estimate the following: y(1) ≈ and y(−1) ≈ (b) For the solution that satisfies y(0) = 1, sketch the solution curve and estimate the following: y(0.5) ≈ and y(−1) ≈ (c) For the solution that satisfies y(0) = −1, sketch the solution curve and estimate the following: y(1) ≈ and y(−1) ≈
Consider the direction field below for a differential equation. Use the graph to find the equilibrium solutions. Answer (separate by commas): y =
Consider the two slope fields shown, in figures 1 and 2 below. figure 1 figure 2 On a print-out of these slope fields, sketch for each three solution curves to the differential equations that generated them. Then complete the following statements: For the slope field in figure 1 , a solution passing through the point (3, −1) has a slope. For the slope field in figure 1, a solution passing through the point (−2, −1) has a slope. For the slope field in figure 2, a solution passing through the point (1, 0) has a slope. For the slope field in figure 2, a solution passing through the point (0, −2) has a slope.
(Section 4.6) Use Euler's method to solve dB dt = 0.06B with initial value B = 1300 when t = 0. A. Δt = 1 and 1 step: B(1) ≈ B. Δt = 0.5 and 2 steps: B(1) ≈ C. Δt = 0.25 and 4 steps: B(1) ≈