Consider the differential equation x°y" + 12x3y" + 25xy' – 25y = 0; x, x, x In(x), (0, 0). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. The functions satisfy the differential equation and are linearly independent since W(x, x5, x In(x)) = * 0 for 0

Consider the function of two variables  u (x,t) = 2e-t sin (2x)  Select the option that gives a partial differential equation for which this function of two variables is a solution.  A. (∂2u/∂x2) = 4(∂2u/∂t2)  B. (∂2u/∂x2) = (1/4)(∂2u/∂t2)  C. (∂2u/∂x2) = - (1/4)(∂2u/∂t2)  D. (∂2u/∂x2) = -4 (∂u/∂t)  E. (∂2u/∂x2) = 4 (∂u/∂t)

Use the variable conversion method to find the general solution of the given differential equation

Consider the differential equation: dy/dx = y(3 - y) 1) Use the slope field plotter to plot different solutions corresponding to A: (0,-2); B: (0,1); C: (0,3); D: (0,4). You can drag each point to the correct place if they're somewhere else on the plane, or click on the point and redefine. Capture a screenshot showing all the solution curves and insert below. f'(x)=y(3-y) Kin/5 -5 Density Length dy de -8 Step size 0.1 Ymas y (3-9) Input. ✔Solution A Solution B Solution C Solution D DCQ= AC: 2) To what value does y(x) approach, as x → ∞, for the solutions corresponding to B,C, and D? To what value does y(x) approach, as x →∞, for the solution corresponding to A? Why is it different from the others? Be specific and answer in terms of what you know about the slope field and its effect on the solution. - 3) Use wolframalpha.com or something similar to find the particular solution corresponding to each of the points A,B,C,D. How is the solution through Point C different from the other…