Question 5: Show that the nontrivial solution of the equation xy" +y+y=0 Satisfying the conditions y'(1) = 0, y' (e) = 0 form an orthogonal set on the indicated interval. Solve the equation, impose the given boundary conditions, obtain the characteristic funagons, and verify their orthogonality by direct integration

Find a solution to the equation (This is an Euler-Cauchy equation)

Write parametric equations for a(t) at (1,1) a(t) = <1, t, 1/(e+t^3)>

The goal of this problem is to fit a trigonometric function of the form f(t) = a sin(t) + b to the data points (0, -6). (,-7). (π,-6). (3, 3), using least squares. (a) The problem is equivalent to finding the least squares solution to the system Xc = y where X= (b) Find the coefficients of the best fit by finding the least squares solution to the system in part (a) a and c = [a, b] b

Consider the Riccatti equation v'=v2-(7/x)v+8/x2 a) use the substitution v=-y'/y b) find the general solution of the resulting second order equation c) Use the information from b) to  find one particular solution v1 of (1).Then use the substitution v = w + v1 to solve (1).

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Euler Equation: t2y''+ty'+y=0, t>0

Explain why a cubic Bézier curve is completely determined by x(0), x (0), х(1), and x'(1).

Consider the Riccatti equation Please answer all parts. v'=v2-(7/x)v+8/x2 a) use the substitution v= -(y'/y) b) find the general solution of the resulting second-order equation c) Use the information from b) to find one particular solution v1 of (1).Then use the substitution v = w + v1 to solve (1).