(10 points) Convert the following high order equation into a first order system, where f is...
differential equation
Convert the IVP into an IVP for a system in normal ( canonical) form: y(+y(O-340=t; x0- 3; y(o = -6 a) b.) Given F(s)= - . Find (f f)( dv= J Solve the integral equation: c) Solve the IVP using Laplace transforms: d.) ty+y-y-O,XO) = 0; y(0) =1
Convert the IVP into an IVP for a system in normal ( canonical) form: y(+y(O-340=t; x0- 3; y(o = -6 a) b.) Given F(s)= - . Find (f f)( dv=...
(2) Convert the following fourth-order ODE to a system of (four) first order ODEs. y(4) + 3y" + 6y" – 9y - 12y = 0 System for (2):
(1) For the following system of ODES: (i) First, convert the system into a matrix equation, then, (ii) Find the eigenvalues, 11 and 12, then, (iii) Find the corresponding eigenvectors, x(1) and x(2), and finally, (iv) Give the general solution (in vector form), ygen, of the system. (Parts (i)-(iii) will be in your work) s y = -241 + 742 y2 = yı + 4y2 General Solution:
A first order linear equation in the form y p(x)y = f(x) can be solved by finding an integrating factor u(x) = exp c) dx (1) Given the equation y 2xy = 10x find H(x) = (2) Then find an explicit general solution with arbitrary constant C у %3 (3) Then solve the initial value problem with y(0) = 3
A first order linear equation in the form y p(x)y = f(x) can be solved by finding an integrating factor...
A first order linear equation in the form y' + pay = f() can be solved by finding an integrating factor H(x) = exp() P(a) dx) (1) Given the equation xy' + (1 + 5x) y = 8e 5 sin(4x) find () = (2) Then find an explicit general solution with arbitrary constant C. y = (3) Then solve the initial value problem with y(1) = e-5
Find a first-order system of ordinary differential equations
equivalent to the second-order nonlinear ordinary differential
equation y ^-- = 3y 0 + (y 3 − y)
(3 points) Find a first-order system of ordinary differential equations equivalent to the second-order nonlinear ordinary differential equation y" = 3y' +(y3 – y).
Exercise 4: (5 points) consider the following differential equation 3y - y Let = f(ty) be the right-hand side of the above equation. a. Compute a/ay. b. Determine and sketch the region in the ty-plane where functions. and array are both continuous C. For the initial condition y(0) = 1 (i.e.to = 0, y = 1), would a unique solution of the equation exist? Explain.
2) Just as an nth order equation can be transformed to a system of n first order equations, a system of m n" order equations can be transformed into a system of mn first order equations. Transform the system of two second order equations into a system of first order equations. Again, your variables are x,, X2, X3..... x" + 3x' + 4x – 2y=0, y" + 2y' – 3x + y =é cost, where x = x(t), y =...
Consider the first order separable equation y(1 + 53*) 1/3 An implicit general solution can be written in the form yCf(x) for some function f(x) with an arbitrary constant. Here f(x) Next find the explicit solution of the initial value problem y(0) = 3 y =
Consider the autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points). f(y) to determine where solutions are increasing / decreasing. Use the sign of y' e) (3 points) Sketch several solution curves in each region determined by the critical poins in the ty-plane
Consider the autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points)....