(2) Consider the system of equationsd2xdt2 + d2ydt2 − 2dxdt = 4d2xdt2 + dydt − 2x + 5y = 11 sin t.(a) Is the system linear or nonlinear?(b) Is the system homogeneous or nonhomogeneous?(c) Convert the given system into an equivalent system of first order equationsand write your answer in the formx′(t) = Ax(t) + b(t)
Question
(2) Consider the system of equationsd2xdt2 + d2ydt2 − 2dxdt = 4d2xdt2 + dydt − 2x + 5y = 11 sin t.(a) Is the system linear or nonlinear?(b) Is the system homogeneous or nonhomogeneous?(c) Convert the given system into an equivalent system of first order equationsand write your answer in the formx′(t) = Ax(t) + b(t)
Solution
(a) The system is linear. This is because all the terms are either constants, the function itself, or its derivatives, each raised to the first power. There are no terms that are products of the function or its derivatives, which would make the system nonlinear.
(b) The system is nonhomogeneous. This is because there are terms on the right-hand side of the equations that are not zero. If the right-hand side of both equations were zero, the system would be homogeneous.
(c) To convert the system into an equivalent system of first order equations, we can introduce new variables for the first derivatives of x and y. Let's denote u = dx/dt and v = dy/dt. Then, the system of equations becomes:
du/dt = -v + 2u + 4 dv/dt = -5v + 2x - u + 11sin(t)
This can be written in matrix form as:
x'(t) = Ax(t) + b(t)
where x'(t) = [du/dt, dv/dt], x(t) = [u, v], A = [[2, -1], [2, -5]], and b(t) = [4, 11sin(t)].
Similar Questions
Consider the following linear system of first order differential equations for x(t)and y(t)x′ = 4x + yy′ = 2x + 5y(a) Write the system using matrix-vector notation.[2 marks](b) Find the characteristic equation, eigenvalues, and eigenvectors of thematrix related to the given system of equations.[6 marks](c) Write the general solution of the system.[4 marks](d) Calculate the fundamental matrix which at t = 0 is equal to the 2 × 2identity matrix.[4 marks](e) Characterize the equilibrium point (0, 0) and plot the phase portrait.
Consider the second order differential equation for x(t)d2xdt2 + adxdt + bx = 0,where a and b are constant real numbers.(a) Convert this differential equation to a two dimensional system of first-order differential equations and then write the system in matrix form.[5 marks](b) If a = 1 and b = −2 find the eigenvalues and the eigenvectors of thematrix associated to the system and write the general solution of thesystem. [10 marks](c) For the same values of a and b characterise the type of the equilibriumpoint at the origin and sketch the phase plane of the system
Find dydx for the following cases : 6(i) 2 3 4[ ( sin ) ]y x x x= + +(ii) 4 4 16x y+ =3. (a) Evaluate the following integrals : 6(i) 2 2921(2 1)t t t dtt+ −∫(ii) 3 / 20 sin x dxπ∫(b) Find dydx , when x xy x xe= + . 44. (a) Find all the roots α, β, γ of the cubicequation 3 7 6 0x x− − = . Also, find theequation whose roots are α + β, β + γand α + γ. 5
Suppose 4x2 + 16y2 = 100, where x and y are functions of t.(a)If dydt = 14, find dxdt when x = 3 and y = 2.dxdt = (b)If dxdt = 4, find dydt when x = −3 and y = 2.
Identify the type of the differential equation (as linear, homogeneous,exact, separable or Bernoulli) and solve the initial value problemdydt = 2 sin(t) − y, with y(π2)= 3
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.