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- Consider the linear system y' = (a) Find the general solution. Ay with 4 - ( -5₁ 4 ) AWrite the given second order equation as its equivalent system of first order equations. u" + 4.5u' – 7.5u = -7 sin(3t), u(1) = -7.5, u'(1) = -6 Use v to represent the "velocity function", i.e. v = u'(t). Use v and u for the two functions, rather than u(t) and v(t). (The latter confuses webwork. Functions like sin(t) are ok.) u' = %3D Now write the system using matrices: u d + dt V V and the initial value for the vector valued function is: u(1) v(1)Find two linearly independent solutions of y''+2xy = 0 of the form in the image
- The number of defects on the front side (X) of a wooden panel and the number of defects on the rear side (Y) of the panel are under study. Suppose that the joint pmf of X and Y is modeled as fxy (x,y)=c(x+y), x=1,2,3 and y=1,2,3. Check if the number of defects on the front side (X) of a wooden panel and the number of defects on the rear side (Y) of the panel are independent.3 Solve the nonhomogeneous system tY': = 2 (-2) ₁ Y+ (1-2₁).Match each linear system with one of the phase plane direction fields. (The blue lines are the arrow shafts, and the black dots are the arrow tips.) ? ✓ | 1. z ' = || a' ? 2. ': = ? 3.' = 4. a: = 11 8] -10 3 1 5 -2 1 -5 -13 10] -10 x2 A x2 с x1 (x2 B 2x2/ D Note: To solve this problem, you only need to compute eigenvalues. In fact, it is enough to just compute whether the eigenvalues are real or complex and positive or negative.
- If e* and xex Are soluhions linearly independent? write the soluhion. are solutions of y"tay'tby-0 AlsoThe equations xy? + 5xzcos(u) + ye" = 10 and x³yz + xe" – 3u²v² = 9 are solved for u and v as functions of x, y and z near the point P where (x,y,z)=(1,1,1) and (u, v) = (5 ,0). Find ()zy at P.4. If x - xy + y' = 1, find y' %3D 3x2 a.): x-3y? 3x2-1 b.) 1-3y C.) 2-3 3y-x 3x²+3y d.)
- 2) Consider the predator-prey model 2 x'= ax -x²-bxy 2 y'=cy - cy² + xy where all the parameters are positive, x is the prey population, y is the predator population and they both grow logistically. Find all equilibria of the model and discuss their stability., Find two linearly independent solutions of 2a?y" – xy' + (2x + 1)y = 0, x > 0 of the form Y1 = x": (1+ a1x + azx² + a3x³ +.) Y2 = x" (1 + b1 + b,x² + bzx³ +.) where ri > r2. Enter a1 a2 az = bị b2 b3 %3D || || ||a.) Form the complimentary solution to the homogeneous equation. y_c(t) = c_1 [ _ _] + c_2 [ _ _] b.) Construct a particular solution by assuming the form y_P(t) = e^(-3t)a and solving for the undetermined constant vector a. y_P(t) = [ _ _] c.) Form the general solution y(t) = y_c(t) + y_P(t) and impose the initial condition to obtain the solution of the initial value problem. [y_1(t) y_2(t)] = [ _ _]