2. (a) Prove that if x1,...,x, are a fundamental set of solutions of the linear homoge- neous system x' = A(t)x then for any c1,..., Cn E R, %3D X = C1X1 + + CnXn is also a solution. (It is not required here that the solutions form an FSS.) (b) Now prove that if y is a solution of the non-homogeneous system x' = A(t)x+g(t) then x.+ y is also a solution. %3D (c) Finally prove that any solution of the non-homogeneous system must be of the form x. +y for some c1, , Cn

Q: (a) Show from first principles, i.e., by using the definition of linear inde- pendence, that if µ =…

A:

Q: Let x(t) = (x1(t), x2(t)) be the unique solution of the system: 2 = a} – a x2 = 1- x122 such that…

A: x1ι=x12-x22x2ι=1- x1x2x1(0)=2 and x2(0)=1Hence at  (2,1)x1ι=4-1=3x2ι=1-2=-1

Q: Prove that the following homogeneous system has only the trivial solution: x+2y + 3z = 0; 2x +5y +…

A: The given system is, x+2y+3z=0    ....12x+5y+3z=0  .....2x+8z=0           ......3

Q: 9. (a) Show that the functions r(t) = 3et+ 7e-t and y(t) = 3et+ 21e are solutions of the system I' =…

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Q: Let X = X(t) be a fundamental matrix of the linear homogeneous system x' = Ax of differential…

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Q: Pls explain to me step by step. pls dont skip any steps. thanks

A: To analyze the given non-linear system using the relevant theorems

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Q: 13.Let x = x1(t), y = y1(t) and x = x2(t), y = y2(t) be any two solutions of the linear…

A:

Q: 2. For the following dynamical system [o oj x'(t) = |x(t) + | x(t) · 2 0 Compute x(0) when u(t) = 0…

A: We have the dynamical system X'(t)=0020X(t)+10U(t)we have to find X(0) when U(t)=0 and X(2)=10now we…

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Q: 1. Consider the following second order linear equation with initial conditions (IVP): e'y" + y – y =…

A: Linear equation can be represented as,                                                         x→' =…

Q: 9. (a) Show that the functions r(t) = 3et+ 7e-t and y(t) = 3et+21e-t are solutions of the system I'…

A: Given: 9. a) x(t)=3et+7e-t and yt=3et+21e-t and a system, x'=2x-yy'=3x-2y satisfying the initial…

Q: Consider the initial value problem of a system of linear differential equations x' = Ax+eAt "A], =…

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Q: 1. (a) Find a general solution of the following linear system of differential equa- tions. I = 2x1 -…

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Q: 2. Find the criticai points of the following system and show that the system is almost linear in the…

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Q: Prove: If xp is a particular solution to the nonhomogeneous system ax=b then every solution ax=b can…

A: xp is a particular solution to the nonhomogeneous system ax=b, therefore, Let x be any solution of…

Q: 2.3 Consider the nonlinear system 1 i = y + x (x2 + y2 –- 1) sin x2 + y2 – 1 1 y = -x+ y (x2 + y2 –…

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Q: 14-Consider the Lie nard equation - (u-x²)x + x = 0. The Jacobian matrix for the linearized system…

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Q: 1. Consider the linear homogeneous system X'(t) = A(t)X(t) for t >1 with 2t2 +1 3 t3 – t 1- t2 A = 1…

A: Given linear homogeneous system  X't=AtXt   for  t>1  with  A=2t2+1t3-t31-t21t2-11t-t3⇒A=2t2+1-t…

Q: 12.Let x = ϕ(t) be the general solution of x′ = P(t)x + g(t), and let x = v(t) be some particular…

A: Given: x = ϕ(t) be the general solution of x′ = P(t)x + g(t), and let x = v(t) be some particular…

Q: Example Consider the first-order system Sy₁ = −2ty₁ + 3y²/ y₂ = − 3y (1 - y2) with the initial…

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Q: 4. Suppose that the matrix P has the following eigenpairs: d1 = 4, v1 = A2 = -5, v2 = Write the…

A: The general solution of given system is given as follows :

Q: 1. Consider linear system R°: ax + by + cz = d ex + fy + gz = h (a) Write down the augmented matrix…

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Q: Consider the initial value problem (x + 2) y" + with initial condition x 1 5Y' + e²y=x³ y (0) = 7 y'…

A: Given the Initial Value Problem x+2y"+1x-5y'+exy=x3 with given initial conditions. y0=7,y'0=2

Q: (b) Compute el for 1 -1 1 Solve the following linear nonhomogeneous system: x = x1 – x2 + 4 x½ = x1+…

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Q: dx/dt=4x(1-2x)-xy dy/dt=2y-4xy (a) Find all critical (equilibrium) points. (b) Using the Jacobian…

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Q: p = jp\p| (j=v-1). Write this as a system in R2 and find a nontrivial periodic solution.

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Q: Let A = Consider the discrete dynamical system x+1 = Axg. If xo = what is x,? Enter it into x_2.…

A: Given discrete dynamical system is xi+1=Axi where matrix A is A=320073

Q: Recall from (14) in Section 8.3 that X = D(t)O¯(t,)x, + D(t) o-(s)F(s) ds solves the initial value…

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Q: Consider the system Sr = 2x2 | = -2x1 Subject to the initial conditions r1 (0) = 3, x2(0) = 4. (a)…

A: Introduction: A system of a differential equation can be converted into a single differential…

Q: Find the general solution in terms of real functions. (b) From the roots of the characteristic…

A: Given,           y''-4y'+4=0

Q: Consider the vectors x1 and (a) Compute the Wronskian of x1 and x2. (b) In what intervals are x1 and…

A: Given: x1=t22t, x2=etet (a) To compute the Wronskian of x1 & x2 (b) To find the intervals where…

Q: Find the solution to the following partially decoupled system with the given { 2r- Sy2 with Y(0) =…

A: Given - dxdt = 2x - 8y2 , dydt = -3y with Y0 = x0, y0 = 0, 1 To find - Find the solution to the…

Q: Consider the following. x1'  =  3x1 − 2x2,      x1(0)  =  3 x2'  =  2x1 − 2x2,   x2(0)  =  1 2 (a)…

A:

Q: Consider a linear system: x`= ux + y, y`= -x + y, where u is a real constant. When is (0,0) an…

A:

Q: 1 1 6. 2 A = -3 -1 1 -2 1 -1 1 4 1 6. 1 3 If the system Ax = b is consistent where b find the…

A: The solution is given as

Q: What are the roots of the characteristic equation of the following system? dx = x – y + 4z dt dy 3x…

A: We find the roots of the characteristic equation below

Q: What are the roots of the characteristic equation of the following system? dr I - y + 4z dt dy 3x +…

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Q: Prove that the general solution of the nonhomogeneous linear system, 6 x-(1-1)×(1)²² · (-)-(-) X + +…

A: Given system of linear differential equation, X'=-1-1-11X+11t2+6-8t+-17 We are provided with…

Q: Solve and find the response of a linear second-order system described by (dy^(2))/(dt^(2))+3…

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Q: 2. Consider the nonhomogeneous system Ax = b given by a1 + x2 = 3 2.x1 – x2 = 3 *1 – 2x2 = 1 Show…

A:

Q: Consider a system of differential equations X Ax, where x = and the 2x2 matrix A has eigenpairs d1 =…

A: To find the given options

Q: pls explain to me step by step and dont skip any step. i will rate your answer immediately. thanks a…

A: To analyze the given non-linear system at its critical point(s)

Q: 7. Sketch the phase-plane behavior of the following systems of linear equations and classify the…

A: Answer is mentioned below As per the guidelines we are suppose to answer only first 3 subparts.…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.