Write the following regarding solutions of a system → x′= A → x: 1. The deﬁnition...

Write the following regarding solutions of a system
→ x′= A → x: 1. The deﬁnition of a Fundamental Matrix Φ(t), and apply it in one example of your choosing (for a constant matrix A that you pick). 2. On your example in part 1. calculate the function y(t) = detΦ(t) (the determinant of Φ(t) and check that y(t) solves the ﬁrst order ODE y′(t) = (trA)y(t) (the trace (trA) is the sum of the entries on the main diagonal of A). 3. Choose a constant 2×2 matrix A different from the one used before. By solving the ODE y′(t) = (trA)y(t) conclude that y(t) is either identically zero, or is never zero.

Expert Solution

Colby Messinger answered 1 year ago

Two linearly independent solutions of the following equation (1 − x) y″ + x y′ − ...

Two linearly independent solutions of the following equation (1 − x) y″ + x y′ − y = 0 are y1(x) = 4ex and y2(x) = 8x. (a) Find the Wronskian W(y1, y2) of y1 and y2. (b) Using the method of variation of parameters, find a particular solution of (1 − x) y″ + x y′ − y = 2(x − 1)2 e −x

Find all solutions to the following equations: (a) √2x − 2 = √x + 1 (b)...

Find all solutions to the following equations: (a) √2x − 2 = √x + 1 (b) x4 − 5x2 + 6 = 0 (c) |3x − 7| < 5 (d) |ax + b| ≥ c Explain step by step please Suppose 4 is a right triangle with leg-lengths a and b and hypotenuse c. Find the missing side: (a) a = 3, b = 4, c =? (b) a = 12, c = 13, b =? (c) a = 6,...

1. Write the acid dissociation reaction for following solutions, and calculate the pH for each of...

1. Write the acid dissociation reaction for following solutions, and calculate the pH for each of them: a) 0.001 M HCl b) 0.005 M H2SO4 c) 0.005 M KOH

1. Find general solutions of the following systems using undetermined coefficients. a.) x′ = −5x +...

1. Find general solutions of the following systems using undetermined coefficients. a.) x′ = −5x + 6y + 1 y′ = −7y + t . b.) x′ = 6x − 5y + e^5t y′ = x + 4y . c.) x′ = −6x − 3y + te^2t y′ = 4x + y .

Write the augmented matrix of the system and use it to solve the system. z. x...

Write the augmented matrix of the system and use it to solve the system. z. x − 2 y − z = − 6 x − y − 2 z = − 4 − x − y + 2 z = − 4

1) Calculate of pH of each of the following strong acid solutions: (a) 8.5 x 10^-3...

1) Calculate of pH of each of the following strong acid solutions: (a) 8.5 x 10^-3 M HBr, (b) 1.52 g of HNO3 in 575 mL of solution, (c) 5.00 mL of 0.250 M HClO4 diluted to 50.0 mL, (d) a solution formed by mixing 10.0 mL of 0.100 M HBr with 20.0 mL of 0.200 M HCl. 2) Calculate [OH-] and pH for (a) 1.5 x 10^-3 M Sr(OH)2, (b) 2.250 g of LiOH in 250.0 mL of solution,...

Write the augmented matrix for the following system of linear equations. Solve the equation (x,y,z) by...

Write the augmented matrix for the following system of linear equations. Solve the equation (x,y,z) by row reduction. x+3y-2z=5 3x+5y+6z=7 2x+4y+3z=8

8. Find the solutions of the following initial value problems • x ''' + 6x ''...

8. Find the solutions of the following initial value problems • x ''' + 6x '' + 12x ' + 8x = 0, x(1) = −2, x' (1) = x '' (1) = 0 • x ^(5) + 4x^ (4) + 5x^ (3) = 0, x(0) = x ' (0) = x '' (0) = 1, x^(3) (0) = x^ (4) (0) =0

Consider the system; dx/dt=x(1-x/10)-1/2xy dy/dt=2y(1-y/6)-xy (a) Find all equilibrium solutions. (b) Draw the phase plane. (c)...

Consider the system; dx/dt=x(1-x/10)-1/2xy dy/dt=2y(1-y/6)-xy (a) Find all equilibrium solutions. (b) Draw the phase plane. (c) On the phase plane, mark • all equilibrium solutions. • the lines where dx/dt =0, dy/dt =0 • an arrow indicating the general direction of the flow of the system in each region on the picture. (d) If you were given initial conditions x = 3 and y = 4, what do you think would happen to the solution?

1. Evaluate the general solutions of the following differential equations: (A) y''+y= 1/cos(x) (B) y'''-y''= -3x+1...

1. Evaluate the general solutions of the following differential equations: (A) y''+y= 1/cos(x) (B) y'''-y''= -3x+1 (C) 2(y')^2= y''(y-1)

Question