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- As illustrated in the accompanying figure, suppose that a rod with one end fixed at the pole of a polar coordinate system rotates counterclockwise at the constant rate of 1 rad/s. At time t = 0 a bugon the rodis 10 mm from the pole and is moving outward along the rod at the constant speed of 2 mm/s.(a) Find an equation of the form r = f(θ)forthe path of motion of the bug,assuming that θ = 0when t = 0.(b) Find the distance the bug travels along the path in part (a) during the first 5 s.Round your answer to the nearest tenth of a millimeter.arrow_forwardClassify each of the following equations as linear or nonlinear (explain you're the reason). If the equation is linear, determine further whether it is homogeneous or nonhomogeneous. a. (cosx)y"-siny'+(sinx)y-cos x=0 b. 8ty"-6t²y'+4ty-3t²-0 c. sin(x²)y"-(cosx)y'+x²y = y'-3 d. y"+5xy'-3y = cosy 2. Verify using the principle of Superposition that the following pairs of functions y₁(x) and y2(x) are solutions to the corresponding differential equation. a. e-2x and e-3x y" + 5y' +6y=0 3. Determine whether the following pairs of functions are linearly dependent or linearly independent. a. fi(x) = ex and f(x) = 3e³x b. fi(x) ex and f2 (x) = 3e* 4. If y(x)=e³x and y2(x)=xe³x are solutions to y" - 6y' +9y = 0, what is the general solution? Question 1. Classify each of the following equations as linear or nonlinear (explain you're the reason). If the equation is linear, determine further whether it is homogeneous or nonhomogeneous. a. (cosx)y"-siny'+(sinx)y-cos…arrow_forwardHELP ASAParrow_forward
- Problem 3: Consider the functional: I = /1+yr2 dx . y a) Find the Euler-Lagrange equation, which minimizes that functional. b) Solve the Euler-Lagrange and show that the curve is a circle. c) Find the radius and the center of the circle.arrow_forwardThis problem is an example of critically damped harmonic motion. A hollow steel ball weighing 4 pounds is suspended from a spring. This stretches the spring feet. The ball is started in motion from the equilibrium position with a downward velocity of 5 feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second). Suppose that after t seconds the ball is y feet below its rest position. Find y in terms of t. Take as the gravitational acceleration 32 feet per second per second. (Note that the positive y direction is down in this problem.) y = learrow_forwardsolve question 4 asap with explanationarrow_forward
- Suppose point T is between points R and V on a Mine. If RT = 6.3 units and RV = 13.1 units, then %3D %3D what is TV? A. 2.5 units B. 6.8 units C. 7.8 units D. 19.4 unitsarrow_forwardQUESTION 8 On some highways, a car can legally travel 20 km/hr faster than a truck. Travelling at maximum legal speeds, a car can travel 120 kms in 18 minutes less than a truck. What are the maximum legal speeds for cars and for trucks? Car = 110 km/hr & Truck = 90 km/hr Car = 120 km/hr & Truck = 100 km/hr Car = 100 km/hr & Truck = 80 km/hr %3D O Car = 90 km/hr & Truck = 70 km/hrarrow_forward1. A space-ship is heading towards a planet, following the trajectory, r(t) = (Ae-¹² cos(3t), √2Ae-t² sin(3t), - Ae-t² cos(3t)), where A 50, 000km and the time is given in hours. (a) The planet is centred at the origin and has a radius, rp = 2,000km. At what time does the ship reach the planet? Give your answer (in hours) both as an exact expression and as a decimal correct to 4 significant figures. (b) To 4 significant figures and including units, what are the velocity and speed of the space-ship when it reaches the planet?arrow_forward
- part d, e, farrow_forwardQuèstion 12 The line that passes through the points (2,0), (0,1) is x+2y = c, where c = 1. True O False A Moving to another question will save this response. ASUSarrow_forwardThe motion of a particle is defined by the following equations: dx/dt = x-2y and dy/dt = 5x-y with x(0) = 2 and y(0) = -1 Find x(t) and y(t) If x(t) and y(t) are periodic, find their amplitudes and periods Which of the attached graphs represents the motion? (Circle answer) A B Neitherarrow_forward
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning