2015 Paper (5 marks) 5. Third order system Solve the differential equation x'" + 14x" +...
Question 5. (4 marks) Consider the first order differential equation y' = x² + y2 subject to the condition y(0) = 0. As discussed in lectures, the solution to this problem for x > 0 has a vertical asymptote. Use the transformation Y u to transform the above differential equation into a second-order linear homogeneous equation. Determine equivalent initial conditions for this transformed equation, and identify what the transformation implies about solutions to the original equation, y.
2. a) Find the solutions (t) and y(t) of the system of differential equations: 10y, y10 by converting the system into a single second order differential equation, then solve it. The initial conditions are given by r(0) 3 and y(0)-4. Show your full work. [7 marks] b) For t = [0, 2n/5]: identify the parametric curve r(t) (t),(t)), find its cartesian equation, then sketch it. Hint: You can use parametric plots in Matlab or just sketch the curve by hand....
3. a) Find the solution y(t) of the ordinary differential equation with the initial conditions: (Solve it only by hand and show your complete work. Do not use a calculator or any symbolic calu lations). [8 marks b) ) Recast our third order ODE into a system of irst order ODEs of the form A.v, where v' = dv/dr = f(v) and v = (y,y,,y")" . You should show all working to find the corresponding matrix A. Do not solve...
4. [10 marks] A second order ordinary differential equation is defined on an interval [0,5) with boundary conditions, and is given as follows 2 + 3ty = 1+ cos(it), y(0) = 1, y(5) = 0 To solve the equation numerically we approximate it on a one-dimensional discrete mesh with N + 1 grid points. That is, we divide the interval (0,5) into subintervals of size h = 5/N and denote t; = ih, y(t) = y(ih)=yi, i = 0,1,... N...
1) Derive the 2d order differential equation for the circuit and solve the equation for a natural response and a forced response using initial conditions. Do not use Laplace Transforms. After finding the differential equation, classify the system as critically damped, overdamped, or underdamped and derive the response equation. 12 V 20㏀ 10 mH
a) Consider the first-order differential equation (y + cos.r) dx + dy = 0. By multiplying integrating factor y(x) = ei" to both sides, show that the differential equation is exact. Hence, solve the differential equation. (6 marks) b) Solve the differential equation (4.r + 5)2 + ytan z = dc COSC (7 marks)
Exact Solution of 1st-order system of Differential Equations Find the Particular solution of the following differential equation with the initial conditions: pls don't solve this using matrices. ー-3-2y, x(t = 0) = 3; 5x - 4y
Solve the given third-order differential equation by variation of parameters. y" - 5y" – y' + 5y = e** y(x) =
Solve the given third-order differential equation using variation of parameters /r
please write the code for the plot Solve the following second order differential equation analytically for x(t): - dx + 5x = 8 * 2 for the following two cases: Case 1: all initial conditions are zero. Case 2: given the initial conditions: x (0) = 1 (0) = 2 For both cases, also plot the solution obtained, for t = 0 to 10.