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Problem 1 A and B
Problem 1. For each differential equation below, rewrite it in operator...
(8 pts) In this problem you will solve the non-homogeneous differential equation y" + 9y =...
(8 pts) In this problem you will solve the non-homogeneous differential equation y" + 9y = sec (3x) (1) Let C and C2 be arbitrary constants. The general solution to the related homogeneous differential equation y" + 9y = 0 is the function yn (x) = C1 yı(2) + C2 y2(x) = C1 +C2 NOTE: The order in which you enter the answers is important; that is, Cif(x) + C2g(x) + C19(x) + C2 f(x). (2) The particular solution yp(x)...
(1 point) Given the fourth order homogeneous constant coefficient equation y' + 10y" + 9y =...
(1 point) Given the fourth order homogeneous constant coefficient equation y' + 10y" + 9y = 0 1) the auxiliary equation is ar4 + bp3 + cp2 + dr te= r^(4)+10r^2+9 0. 2) The roots of the auxiliary equation are 1,-1,31,-3i (enter answers as a comma separated list). 3) A fundamental set of solutions is cosx,sinx,cos(3x), sin(3x) (Enter the fundamental set as a commas separated list 41, 42, 43, 44). Therefore the general solution can be written as y =...
(1 point) We consider the initial value problem 4xy" + 4xy' +9y = 0, y1) =...
(1 point) We consider the initial value problem 4xy" + 4xy' +9y = 0, y1) = 1, y'(1) = -3 By looking for solutions in the form y = x" in an Euler-Cauchy problem Ax?y' + Bxy + Cy = 0, we obtain a auxiliary equation Ar2 + (B – A)r + C = 0 which is the analog of the auxiliary equation in the constant coefficient case. (1) For this problem find the auxiliary equation: = 0 (2) Find...
Given a second order linear homogeneous differential equation a2(x)” + a (x2y + a)(x2y = 0...
Given a second order linear homogeneous differential equation a2(x)” + a (x2y + a)(x2y = 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions yı, y. But there are times when only one function, call it yi, is available and we would like to find a second linearly independent solution. We can find y2 using the method of reduction of order. First, under the necessary assumption the az(x) + 0 we rewrite...
Linear Algebra and differential Equations. Consider the RLC circuit with R = 180Ω, C = 1/280F,...
Linear Algebra and
differential Equations.
Consider the RLC circuit with R = 180Ω, C = 1/280F, L = 20H, and applied voltage E(t) = 10 sin t. Assuming no initial charge on the capacitor, but an initial current of 1 ampere. Determine the charge on the capacitor for t> 0. (a) Write the differential equation, y', +4xy'-6x2y = x2 sin x, as an operator equation and give the associated homogeneous DE (b) Write the DE (D2 1) (D 3)(y) e...
Problem 11 Previous Problem Problem List Next Problem (1 point) Consider the differential equation 2x(x –...
Problem 11 Previous Problem Problem List Next Problem (1 point) Consider the differential equation 2x(x – 1)y" + 3(x - 1)y' - y = 0 which has a regular singular point at x = 0. The indicial equation for x = 0 is p2 + r+ = 0 with roots (in increasing order) rı = and r2 = Find the indicated terms of the following series solutions of the differential equation: (a) y = x" (3+ x+ x2+ x +...
4. Problem 4. Consider the following system of first order coupled ordinary differential equation...
4. Problem 4. Consider the following system of first order coupled ordinary differential equations, where r (t) and a) Rewrite the initial value problem (IVP) in a matrix form aAi, where ? r (0) +v()() b) Find the three distinct (real) eįgrivalus {A] c) Verify that, satisfies the IVP where the constant ακ fficients c1 c2 and C3 can be detennined from the three given initial conditions. P BIVPn initial 5. Problem 5 (challenge problem): Sinultaneous diagonalization of commuting matrices...
Consider the following statements. (i) Given a second-order linear ODE, the method of variation of parameters...
Consider the following statements.
(i) Given a second-order linear ODE, the method of variation of
parameters gives a particular solution in terms of an integral
provided y1 and y2 can be
found.
(ii) The Laplace Transform is an integral transform that turns
the problem of solving constant coefficient ODEs into an algebraic
problem. This transform is particularly useful when it comes to
studying problems arising in applications where the forcing
function in the ODE is piece-wise continuous but not necessarily...
dy: 2 Consider the following Ordinary Differential Equation (ODE) for function yı(z) on interval [0, 1]...
dy: 2 Consider the following Ordinary Differential Equation (ODE) for function yı(z) on interval [0, 1] +(-10,3) dayi dy + 28.06 + (-16.368) + y(x) = 1.272.0.52 with the following initial conditions at point a = 0; dy 91 = 4.572 = 30.6248 = 185.2223 dar Introducting notations dyi dy2 dy dar dar dir? convert the ODE to the system of three first-order ODEs for functions y1, y2, y3 in the form: dy dar fi (1, y1, ya, y) dy2...
Problem 5. (1 point) A Bernoulli differential equation is one of the form +P()y= Q()y" (*)...
Problem 5. (1 point) A Bernoulli differential equation is one of the form +P()y= Q()y" (*) Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u =y- transforms the Bemoulli equation into the linear equation + (1 - x)P(3)u = (1 - x)^(x). Consider the initial value problem ry' +y = -3.xy?, y(1) = 2. (a) This differential equation can be written in the form (*) with P(1) =...
(8 pts) In this problem you will solve the non-homogeneous differential equation y" + 9y = sec (3x) (1) Let C and C2 be arbitrary constants. The general solution to the related homogeneous differential equation y" + 9y = 0 is the function yn (x) = C1 yı(2) + C2 y2(x) = C1 +C2 NOTE: The order in which you enter the answers is important; that is, Cif(x) + C2g(x) + C19(x) + C2 f(x). (2) The particular solution yp(x)...
(1 point) Given the fourth order homogeneous constant coefficient equation y' + 10y" + 9y = 0 1) the auxiliary equation is ar4 + bp3 + cp2 + dr te= r^(4)+10r^2+9 0. 2) The roots of the auxiliary equation are 1,-1,31,-3i (enter answers as a comma separated list). 3) A fundamental set of solutions is cosx,sinx,cos(3x), sin(3x) (Enter the fundamental set as a commas separated list 41, 42, 43, 44). Therefore the general solution can be written as y =...
(1 point) We consider the initial value problem 4xy" + 4xy' +9y = 0, y1) = 1, y'(1) = -3 By looking for solutions in the form y = x" in an Euler-Cauchy problem Ax?y' + Bxy + Cy = 0, we obtain a auxiliary equation Ar2 + (B – A)r + C = 0 which is the analog of the auxiliary equation in the constant coefficient case. (1) For this problem find the auxiliary equation: = 0 (2) Find...
Given a second order linear homogeneous differential equation a2(x)” + a (x2y + a)(x2y = 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions yı, y. But there are times when only one function, call it yi, is available and we would like to find a second linearly independent solution. We can find y2 using the method of reduction of order. First, under the necessary assumption the az(x) + 0 we rewrite...
Linear Algebra and
differential Equations.
Consider the RLC circuit with R = 180Ω, C = 1/280F, L = 20H, and applied voltage E(t) = 10 sin t. Assuming no initial charge on the capacitor, but an initial current of 1 ampere. Determine the charge on the capacitor for t> 0. (a) Write the differential equation, y', +4xy'-6x2y = x2 sin x, as an operator equation and give the associated homogeneous DE (b) Write the DE (D2 1) (D 3)(y) e...
Problem 11 Previous Problem Problem List Next Problem (1 point) Consider the differential equation 2x(x – 1)y" + 3(x - 1)y' - y = 0 which has a regular singular point at x = 0. The indicial equation for x = 0 is p2 + r+ = 0 with roots (in increasing order) rı = and r2 = Find the indicated terms of the following series solutions of the differential equation: (a) y = x" (3+ x+ x2+ x +...
4. Problem 4. Consider the following system of first order coupled ordinary differential equations, where r (t) and a) Rewrite the initial value problem (IVP) in a matrix form aAi, where ? r (0) +v()() b) Find the three distinct (real) eįgrivalus {A] c) Verify that, satisfies the IVP where the constant ακ fficients c1 c2 and C3 can be detennined from the three given initial conditions. P BIVPn initial 5. Problem 5 (challenge problem): Sinultaneous diagonalization of commuting matrices...
Consider the following statements.
(i) Given a second-order linear ODE, the method of variation of
parameters gives a particular solution in terms of an integral
provided y1 and y2 can be
found.
(ii) The Laplace Transform is an integral transform that turns
the problem of solving constant coefficient ODEs into an algebraic
problem. This transform is particularly useful when it comes to
studying problems arising in applications where the forcing
function in the ODE is piece-wise continuous but not necessarily...
dy: 2 Consider the following Ordinary Differential Equation (ODE) for function yı(z) on interval [0, 1] +(-10,3) dayi dy + 28.06 + (-16.368) + y(x) = 1.272.0.52 with the following initial conditions at point a = 0; dy 91 = 4.572 = 30.6248 = 185.2223 dar Introducting notations dyi dy2 dy dar dar dir? convert the ODE to the system of three first-order ODEs for functions y1, y2, y3 in the form: dy dar fi (1, y1, ya, y) dy2...
Problem 5. (1 point) A Bernoulli differential equation is one of the form +P()y= Q()y" (*) Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u =y- transforms the Bemoulli equation into the linear equation + (1 - x)P(3)u = (1 - x)^(x). Consider the initial value problem ry' +y = -3.xy?, y(1) = 2. (a) This differential equation can be written in the form (*) with P(1) =...
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