Please answer the following questions that a person new to this course would be able to understand.( Include theorem.)
Given the linear systems of differential equations
The sketch of the direction field (v_x, v_y) in this case is
a)
We use the method of elimination by taking one more
time derivative of the 1st equation of x(t)
Now in order to eliminate y'(t), we use the 2nd equation
And so,
And from the first equation, we replace y, i.e.,
And this implies
................(1)
This is the second order differential equation.
b)
And let us assume that the ansatz for the solution
is
And so, putting this in the equation (1), we get
So, the two roots are equal. And so, the general solution is
where, A and B are two undetermined constants which are to be fixed
by the initial conditions.
The initial conditions are
And
The first condition
implies
And taking the derivative of the solution x(t), we get
And so, the 2nd initial condition implies
Now given
So, we get
Nw we have already got, A = 1, so, we get
And so, the solution is
And so,
And so, from the first equation,
So, as we have already got an answer for x(t) and x'(t), so, we
get
This is the solution for y(t).
Please answer the following questions that a person new to this course would be able to...
Solve the system of linear equations, using the Gauss-Jordan elimination method. (If there is no solution, enter NO SOLUTION. If there are infinitely many solutions, express your answer in terms of the parameters t and/or s.) x − 2y + 3z = 3 2x + 3y − z = 0 x + 2y − 3z = −7 (x, y, z) = ( )
I tried to do all 3 problems and I am not be able to get. Help. Thanks. dy 3. Given the differential equati . sketch the direction field, using isoclines, and & a few representative solution curves. Include any linear solutions find linear solutions (of the form y mx + b) find the general solution of the equation ( create a new dependent variable w = V. Then find how砮and 응 are related. Then write down & solve a differential...
30 W01 - Apolled Harnwest-prod-01 Use the methods of section 8.2 to find the general solutions of the given systems of differential equations in the following three problems. 1. dx dt = x+y 2. dx = 2x + y dy = 5x - 3y dt dy dt dt = -x + 4y 3. - X - 4y dx dt dy dt = 2x + 3y Use the methods of section 8.3 to find the general solutions of the given systems...
non-homo 2nd order linear equations 1. Find the general solution for each of the following differential equations (10 points each): (a) (b) (e) y" – 2y! - 3y = 3e2x y" — y' – 2y = -2.3 + 4.2? y" + y’ – 67 = 1234 + 12e-2x y" – 2y' – 3y = 3.ce-1 y" + 2y' + y = 2e- (Hint: you'll use Rule 7. at least once) (e 2. Find the solution to the following differential equation...
3. Homogeneous linear systems with complex and repeated eigenvalues. Find the general solu- tion of the given system of differential equations. For the two-dimensional systems, classify the origin in terms of stability and sketch the phase plane (a) x'(t) y'(t) 6х — у, 5х + 2y. = (b) 4 -5 x'(i) х. -4 (c) 1 -1 2 x'() -1 1 0x -1 0 1 3. Homogeneous linear systems with complex and repeated eigenvalues. Find the general solu- tion of the...
Please answer the following questions giving all details 1. An investment company has 55 million dollars to invest this year. Currently, the company is considering different ways to invest the money. Their objective is to have maximum return on the investment. What are the different investment alternatives that may be used to invest the 55 million dollars and what are the important factors in each alternative that should be considered in making the decision? 2. Find the slope, the y-intercept,...
Solve the following questions and Choose the correct answer. 1) The General solution to y" + y = 0 sty -3&y(x) = -3 y = cos(3x) + sin(-31) , 3cos(x) – 3 sin(x) 3 ) 3 Answer 2) Suppose that y(t) and y(t) are two solutions of a certain second order linear differential equation, sin(t)y" + cos(t) y' - y = 0. 0<<< What is the general form of the Wronskian Wy ) (6) ? Without solving the equation. b)...
1. For each of the following systems of linear equations, find: • the augmented matrix • the coefficient matrix • the reduced row echelon form of the augmented matrix • the rank of the augmented matrix • all solutions to the original system of equations Show your work, and use Gauss-Jordan elimination (row reduction) when finding the reduced row echelon forms. (b) 2 + 2x W 2w - 2y - y + y + 3z = 0 = 1 +...
can you please work number 2? Use the methods of section 8.2 to find the general solutions of the given systems of differential equations in the following three problems. dx 1. - x + y 2. = - 2x + y dt dy dx dt dy dt = 5x - 3y = -x + 4y dt 3. =- X - 4y dx dt dy dt = 2x + 3y
may you please show work -/8 POINTS ZILLDIEFEOMOD ZILLDIFFEQMODAP11 7.1.026. f(t)}. (Write your answer as a function of s.) Use Theorem 7.1.1 to find f(t) = (3t - 1)3 ${f(t)} = -/8 POINTS ZILLDIFFEQMODAP11 4.9.002. Solve the given system of differential equations by systematic elimination. dx = 6x + 10y = x - 3y (x(t), y(t)) =(