Consider the following system of second-order differential equations: x ̈ = −0.7x + 0.9y, y ̈ = 0.9x + 1.7y. The coefficient matrix is [-0.7 0.9] [0.9 1.7] and it has eigenvectors [1] [3] and [3] [-1] Calculate the eigenvalues that correspond to the given eigenvectors and then write down the general solution of the given system of second-order differential equations

Consider the following system of second-order differential equations: x ̈ = −0.7x + 0.9y, y ̈ = 0.9x + 1.7y. The coefficient matrix is [-0.7 0.9] [0.9 1.7] and it has eigenvectors [1] [3] and [3] [-1] Calculate the eigenvalues that correspond to the given eigenvectors and then write down the general solution of the given system of second-order differential equations

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Suppose that the differential equation x' = Ax is such that A is a 2 × 2 matrix with eigenvalues A1…

Q: Find a fundamental matrix of the following system, and then apply x(t) = Þ(t)Þ(0)¯¯ 1x to find a…

A: Step 1:Step 2:

Q: You are given the following inhomogeneous system of first-order differential equations for x(t) and…

A: The given system of first-order differential equations, x'=2x+y+3ety'=4x-y We have to write the…

Q: Here we have a system of differential equations. (dx/dt) = 11x + 8y, (dy/dt) = 8x - y. The…

Q: Consider the linear system a. Find the eigenvalues and eigenvectors for the coefficient matrix. 2 X₁…

A: Find eigenvalues of A are given below-: A-λI=0 And then find eigenvectors with the help of,…

Q: Consider the initial value problem i' = y, j(0) = a. Find the eigenvalue A, an eigenvector i1, and a…

Q: Express the following first-order system as a single higher-order differential equation and solve…

Q: 3. Write the system of ODES: x' = x+z y' = y -z in matrix notation. Now, use eigen values and…

Q: d²x, 2 dx, dt dx2 + 12x1 – 2x2 = 0 dt² dt d²xz 3- dt2 1 dx1 5. 5. dt Txp° dt dx2 - 2x, + 2x2 = 0 1…

A: Given, 2d2x1dt2+5dx1dt-5dx2dt+12x1-2x2=03d2x2dt2-5dx1dt+5dx2dt-2x1+2x2=0 Let u1=x1v1=x1˙=u1˙

Q: Consider the following system of differential equations dx + 3x – y = 0, dt dy + 10x – 4y = 0. dt а)…

Q: ving nonlinear system of differential equations: x² - []=[x² 2²50] + y² - 50

Q: Let y' = Ay be a system of differential equations where A = T The matrix has spectrum X(A) = {-3}…

Q: Express the following first-order system as a single higher-order differential equation and solve…

A: NOTE: Refresh your page if you can't see any equations. . the given system is the matrix A is

Q: Evaluate the differential equation dx/dt=Bx (assuming that x is a function of t) by determining the…

A: first we will find the eigenvalues and corresponding eigenvectors of B then by using these…

Q: f 02 3 mere dz zi #-[21]) Z + N N 2 e*3% and Z=

Q: Given below is the second-order system of differential equations: d2x/dt2 = -x - 3y, d2y/dt2 = -3x…

A: Given: Second-order system of differential equations: d2xdt2=-x-3y, d2ydt2=-3x-y…

Q: . A) Find the general real solution of the differential equation (2) (t) – 21 (t)+ a(t) = 0 and…

A: We solve first que only as per the guidelines. If you want other to be solved, post again by…

Q: a. Find the eigenvalues and eigenvectors of the matrix -12 -16 A1 = , V1 = A2 = v2 = 9. b. Solve the…

A: Given matrix To determine Eigen values and Eigen vectors

Q: Given matrix a b C A = 0 d e 0 0 f (a) Find the eigenvalue(s) of A. (b) Find the corresponding…

Q: 2. Consider the system of differential equations. Determine the eigenvalues and corresponding…

Q: Express the given system of higher-order differential equations as a matriİx system in normal form.…

Q: Consider the system of differential equations dx = x + 4y dt (Σ) dy 2x – y dt (i) Write the system…

Question

Consider the following system of second-order differential equations:

x ̈ = −0.7x + 0.9y,
y ̈ = 0.9x + 1.7y.

The coefficient matrix is

[-0.7 0.9]
[0.9 1.7]

and it has eigenvectors

[1]
[3]

and

[3]
[-1]

Calculate the eigenvalues that correspond to the given eigenvectors and then write down the general solution of the given system of second-order differential equations.

With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step by step

Solved in 4 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

Consider the following second-order, nonlinear system of ordinary differential equations, and its initial conditions. A. Linearize the right-hand sides of Eq. (4.1) and Eq. (4.2) about the stated initial conditions and simplify the resulting equations. B. Transform the results in (A) to Laplace space using Laplace transforms and simplify. All X(s) terms and Y(s) terms should be on the left-hand side of both equations and all other terms should be on the right-hand side of both equations. C. Solve for X(s) and Y(s). D. Convert the right-hand-sides of the results in (C) to a form where the individual terms may be transformed back to time space using inverse Laplace transforms. E. Apply inverse Laplace transforms to the result in (D).
Express the system of differential equations in matrix notation. - x" -6x+ty (sint)x = 0 y'"'+5y"-tx' + 3y' + e¯¹x=0 Which of the following sets of definitions allows the given system to be written as an equivalent system in normal form using only the new variables? OA. x₁ = x, x₂ = x², x3 = y, x4 = y' OB. X₁ =x', x=x", x3 = y', x4 = y", x5 =y"" OC. x₁ = x, x2 =x', x3 = y, x4 = y', x5 =y" OD. x₁ = x, x₂ =x", x3 = y, x4 = y'"' Write the system of equations using matrix notation. Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer.) x1 X1 x2 X3 ○ A. ха X5 X5 ہیں ○ B. کا 5 x1 x2 X3 ха
(2 cos 2t / 19. Assume that a real 2 x 2 matrix A has an eigenvalue of A1 = 3+ 2i with a corresponding eigenvector = (6) Find the general solution of the system of differential equations 2i given by 글 %=D Az. cos 2t sin 2t 교(t) = cie3t ( sin 2t + cze³t 2 cos 2t cos 2t sin 2t a(t) = c1e3t + cze3t -2 sin 2t 2 cos 2t (. cos 2t sin 2t a(t) Ciešt + cze3t sin 2t - cos 2t ( -2 cos 2t sin 2t a(t) = cie3t + cze3t sin 2t -2 cos 2t cos 2t ( 2 sin 2t ) sin 2t a(t) = c1e3t + cze³t 2 cos 2t
Q3. (a) If Ax = 2 x, determine the eigenvalues and the corresponding eigenvectors for (1 A 2 Hence, write down the associated modal matrix P and diagonal matrix D, and use these values to solve the following system differential equations: * = X1 + 4 x2 *2 = 2 x1 + 3 x2 %3D Given that whent 0, x, 0 and x2 2. (b) Use the 4th order Runge Kutta method to solve the differential equation: dy e* y %3D dx for values of x = 0 (0.2) 0.4 given that y = 1 when x = 0. Give your answers correct to 5 decimal places. Obtain the analytical solution of the differential equation and compare the analytical solution when x 0.4 with the values obtained using Runge Kutta,
a. Find the eigenvalues and eigenvectors of the matrix x₁ (t) x₂ (t) = X₁ b. Solve the system of differential equations a' = = = V₁ = [₁ 5 -3 4 -2 and X₂ = satisfying the initial conditions V₂ = [₁8] =[3]
Consider the initial value problem: x1(0) = 4, x2(0) = 6. 4x2, — 4т1 + 812, a. Find the eigenvalue A, an eigenvector v, and a generalized eigenvector w for the coefficient matrix of this linear system. b. Find the most general real-valued solution to the linear system of differential equations. Use c1 and c2 to denote arbitrary constants, and enter them as "c1" and "c2". x1(t) = x2(t) c. Solve the original initial value problem. x1(t) 22(t) Instruction ||||
y" + 5y' – 2y = 0. Make the substitution x1 = y' and x2 x' = Ax where A is a 2 x 2 matrix and x = [1, x2]". Find the matrix A. = y to write an equivalent system of first order differential equations of the form
Suppose A is a 2 × 2 real matrix with an eigenvalue λ = 3 + 5i and corresponding eigenvector i * = (-²+1). v= Determine a fundamental set (i.e., linearly independent set) of solutions for ÿ' = Aỹ, where the fundamental set consists entirely of real solutions. Enter your solutions below. Use t as the independent variable in your answers. (t) = 2(t) = → - →
Suppose A is a 2 × 2 real matrix with an eigenvalue λ = 4 + 3¿ and corresponding eigenvector -1 + i] -[-¹+1). i Determine a fundamental set (i.e., linearly independent set) of solutions for ÿ' = Ay, where the fundamental set consists entirely of real solutions. Enter your solutions below. Use t as the independent variable in your answers. ÿ₁(t) = = v = ₂(t) =
You are given the following simultaneous linear differential equations: dx/dt = x dy/dt = x - y 1. Express the system of equations in matrix form 2. Find the eigenvalues and eigenvectors of the matrix of coefficients 3. Hence or otherwise find the general solution of this system