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- 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)] = [ _ _]arrow_forwardThis is the first part of a three-part problem. Consider the system of differential equations y1 - 2y2, y1 + 4y2. Yi Y2 Rewrite the equations in vector form as '(t) = Ay(t). j'(t) = 身arrow_forwardConsider the initial value problem: 2t a'= ()+ a0), a(1) = (-). 2t2 t+a Suppose we know that æ(t) = (a,) is the unique solution to this initial value problem. Find the constants a and B and the vector function g(t). 3t2 + B a = B = g(t) =arrow_forward
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- 4. Given the system dr -6x – y dt dy dt. 3x 2y (a) Find the general solution. (b) Find the solution that satisfies the initial condition X (0) = (1, –1). Report your solution as one vector.arrow_forwardUse Newton's method with the starting point (1,2)T to find a solution of the system +y 0 2+y = 1. The first iteration produces the vector (A) (2,2)T (B) (2,1) (C) (1,1)" (D) (1,2)" (E) none of the above O (A) O (B) O (C) O (D) O (E)arrow_forwardAt time 0, Cheryl deposits X into a bank account that credits interest at an annual effective rate of 6%. At time 3, Gomer deposits 10000 into a different bank account that credits simple interest at an annual rate of y%. At time 5, the annual forces of interest on the two accounts are equal, and Gomer's account has accumulated to Z. Calculate Z.arrow_forward
- the relationship between aphids, A, ( prey) and ladybugs, L, (preditor) can be described as follows: dA/dt=2A=0.01AL dL/dt=-0.5L+0.0001AL a) find the two critical points of the predator-prey equations b) use the chain rule to write dL.dA in terms of L and A c) suppose that at time t=0, there are 1000 aphids and 200 ladybugs, use the Bluffton university slope field generator to graph the slope for the system and the solution curve. let 0 be less than or equal to A who is less than or equal to 15000 and 0 is less than or equal to L which is less than or equal to 400.arrow_forwardConsider the IVP: y'' - 2y' - 3y = 0, y(0) = 0, y'(0) = 1 Find two linearly independent solutions of the DE (give the interval for which the solutions are valid).Then write the general solution of the DE, finally, use this to solve the IVP.arrow_forwardLet z = 8x2 + 7y2, and suppose that (x, y) changes from (2, -1) to (2.01, -0.98). (a) Compute Az. Az = (b) Compute dz. dz = (c) Compare the values of Az and dz. O dz is a good approximation of Az. O dz is not a good approximation of Az.arrow_forward
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