In Problems 1–10 solve Laplace’s equation (1) for a rectangular plate subject to the given boundary conditions.
4.
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- 5. Show how the origin changes stability as a changes by using the Poincaré Diagram. 2 a x' X 1 2arrow_forward1. Determine the signs (+,- or 0) of r'(t) for the space-curve drawn below: r'(1)=<_ r'(2)=<_ r'(3) =< r'(4)=< r'(5) =< X solve applications. T(1) 1=4 1=5 Z 1=6 Fig. 15 1=2arrow_forward4. Determine when the following pairs of functions are linearly independent. (a) yı(t) = erit; y(t) = er²t, r₁,72 € R (b) y(t) = cos(at); 32(t) = sin(at), a = R (c) y₁ (t) = cosh(at); y₂(t) = sinh(at), a € Rarrow_forward
- Problem 4.3 3 Evaluate the line integral ſc F·dr. F = ei + xej + xyeŸ²k. C is the curve x=t, y=t², z=t³ with 0 ≤ t ≤ 2.arrow_forward1. The coordinate of a point undergoing rectilinear motion is given by x(t) = t³ – 4t, -2arrow_forward.Please solve all the partsarrow_forwardProblem 3: Find the second derivative y of In Iny) + Inx= Iny. B X+Y C. D. in y y-x inx x-yarrow_forwardLine Integral Calculus 3arrow_forwardClassify 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. Find P.S. of (16x +5y) dx + (3x+y) dy = 0 If the curve to pass thru the point (1,-3) 2. Find P.S. of y (2x²-xy +y ²) dx = x²(2x-1) dy o when x = 1 y = ½ }arrow_forwardProblem 4.3 5 Evaluate the line integral [c F.dr. F = yi - xj + 3xyk. C consists of the semicircle x²+y²=4, z=0, y> 0 and the line segment from (-2, 0, 0) to (2, 0, 0).arrow_forwardProblem 2: a. Using the chain rule dy dy dr dr dt dt compute 4 using the parametric equations x = cost, y = sin t, te (-0, 00) Write as a function (i) of t. (ii) of æ and y. (Express the derivative as a rational function, not a piecewise-defined function.) b. Use the chain rule to express in terms of d (dy dt dr dr and dt c. Use the result in part (a) and the chain rule again to compute 4 as a function (i) of t. (ii) of r and y. (Express the second derivative as a rational function, not a piecewise-defined function.) d. Find the derivatives and 4, as in the previous parts, for the parametric equations x = cos 3t, y = sin 3t, te (-0, 0). e. More generally, let f(t) and g(t) be differentiable functions defined over t € (-0, 0). Suppose the curve C1 has parametric equations r = f(t), y = 9(t), te(-∞,00) and the curve C, has parametric equations x = f(2t), y = g(2t), te (-0, a0). Compute both of the derivatives 4 for C1 and for C2, then describe the relationship between these derivatives…arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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