Consider a capacitor made of two very large perpendicular plates. (Let the positive
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- 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 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?arrow_forwardPlease solve question 2.arrow_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
- 8. Any cable hanging between two poles must follow a specific curve. To simplify calculations, we choose a coordinate system so that the locations of the two poles are at x = -B and x=+B. The curve has equation y = C +A(e¹/A +e-2/A). Find an integral expression for the length of the cable, and evaluate that integral. (Your answer will be in terms of A and B.)arrow_forwardClassify each of the following equations as linear or nonlinear (explain you’re reason). If theequation is linear, determine further whether it is homogeneous or nonhomogeneous.arrow_forwardSolve question 3.arrow_forward
- Classify each of the equations above as autonomous, separable, linear, homogeneous, exact, Bernoulli, function of a linear combination, or neither. A. y' = sin(x) + cos(y); B. y = +2 C. y = y; D. y' = ² + y² +eª²-y²; E. y' = x³y + y³x; F. y'=sin(x+2y-2) + cos(x +2y + 1); G. y' = 2y H. y' = x³ +y³r; I. (x² + 2xy)dx + (8x² + 5y²)dy = 0; J. y' = x³.arrow_forward1. Convert the following difference equation into a first-order form: Yt = Yt-1 + 2yt-2(Yt-3 – 1)arrow_forwardA classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensio a²u №²u 2 Laplace's equation is Əx u(x,y) = ex cos(-6y) = + = 0. Show that the following function is harmonic; that is, it satisfies Laplace's equ dy² Find the second-order partial derivatives of u(x,y) with respect to x and y, respectively. a²u a²uarrow_forward
- 21. Valeria is playing with her accordion. The length of the accordion A(t) (in cm), after she starts playing as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form A(t) =a•cos (b-t) +d. At t=0, when she starts playing, the accordion is 15 cm long, which is the shortest it gets. 1.5 seconds later the accordion is at its average length of 22 cm. Find A(t). *arrow_forwarda. Solve the following. 1. Find the angle of intersection between the curves y = Vx and y = 2 – x². 2. At what point of the parabola y = x² – 2x + 3 is the tangent line parallel to 4x + y + 7 = 0? Find its equation. 3. Find a and b so that the parabola y = ax² + bx will have a critical point at (2, 4).arrow_forward7. The velocity of an object traveling along a straight line is v(t) = cos(t) + 4t + 1 meters per second. At t = 1 second, it is located 4 meters to the left of a post located at x = 0. (For this problem, leftwards is the negative x direction and rightwards is the positive x direction.)arrow_forward
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