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Change of Coordinates in Two Dimensions Suppose that E is an ellipse centered at the origin. If the major and minor axes are horizontal and vertical, as in figure 15.1, then the equation of the ellipse is (15.1) x2 a2 + y2 b2 = 1; where a and b are the lengths of the major and minor radii. Figure 15.1 x y Figure 15.2 x y u v However, if the axes of E are neither horizontal nor vertical, as in



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Example 2. y0 = 1 + x2 ¡ y2 has the solution y = x and the change of variable y = x + 1=v transforms the equation into v 0 +2 xv = 1. 2 Orthogonal Trajectories.

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Verify that y = xe−2x is a solution of the differential equation d 2y dx2 + 4 dy + 4y = 0. tHinK WritE 1 Use the product rule for differentiation to fi nd the fi rst derivative.

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2 2Xand cy 1 +y 2 2Y, so their sum cv 1 + v 2 is in X+ Y. Thus X+ Y is a subspace. (It is non-empty because both Xand Y are.) 3. Show that if 0 2LˆV, then Lis linearly dependent. Solution. [5 points] The set f0gis linearly dependent because 10 = 0. Thus Lcontains a linearly dependent set, hence by the previous exercise it is linearly dependent. 4. Determine which of the following subsets of

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5 ay by cy 0 P Q R c 1 c 2 y x c 1y 1 x c 2y 2 x 4 y 1 y 2 P x. Thomson Brooks-Cole copyright 2007 SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 3 EXAMPLE 1 Solve the equation . SOLUTION The auxiliary equation is whose roots are , . Therefore, by (8) the general solution of the given differen-tial equation is We could verify that this is indeed a solution by differentiating and …

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