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5. (a) Write an ODE to which (x - 2)e-ot is a solution. (Hint: Think what...
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...
8. (9 points) Suppose the characteristic equation of a certain twentieth order, linear, constant coefficient, homogeneous...
8. (9 points) Suppose the characteristic equation of a certain twentieth order, linear, constant coefficient, homogeneous differential equation has roots: 2,0, a, 2+3i, ti, +4i, ti, 2, 3, a, 2+3i ,2,3,0, and -3. (where a is a real constant) Write the general solution to this differential equation. (Do not attempt to solve for the coefficients).
Problem #2: Consider the following statements. [6 marks) (1) The particular solution of the ODE)" -...
Problem #2: Consider the following statements. [6 marks) (1) The particular solution of the ODE)" - 6y' + 9y = 5e3x is given by yp = Cre3x where C is an undetermined constant. (ii) The procedure of finding series solutions to a homogeneous linear second-order ODEs could be accurately described as the "method of undetermined series coefficients". (iii) Most of the material in Lecture Notes from Week 3 to Week 5, inclusive, can be extended or generalized to higher-order ODES...
1. Second order ODE (25 points) a. Consider the following nonhomogeneous ODEs, find their homogeneous solution,...
1. Second order ODE (25 points) a. Consider the following nonhomogeneous ODEs, find their homogeneous solution, and give the form (no need to determine coefficients) of nonhomogeneous solution. (12 points) i. 44'' + 3y = 4x sin ( *2) ii. J + 2 + 3 = eº cosh(22) b. Find the general solution of y" + 2Dy' + 2D'y = 5Dº cos(Dx) where D is a real constant with following steps i) Determine homogeneous solution, ii) Find nonhomogeneous solution with...
2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a...
2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a vertical plane under the influence of gravity can be modelled with the second order non-homogeneous ODE 0"(t) + 270'(t) +w20(t) = f(t), (2) where wa = g/l. The second term in the equation represents the damping force (e.g. air resistance) for the given constant 7 > 0. The model can be used to approximate the motion of a magnetic pendulum bob being driven by...
(1) Use the "Schoolboy's Trick" to find a system of two ODE (first order homo geneous linear, wit...
(1) Use the "Schoolboy's Trick" to find a system of two ODE (first order homo geneous linear, with constant coefficients) in two "unknowns" r1,r2 which is "equivalent" to the second order homogeneous linear ODE with constant in the sense that establishes a bijection (one-to-one correspondence... in fact is is an "isomor- phism of vector spaces") from the set of solutions to the above ODE to the set of solutions to your system. Write your answer in the form Ar for...
Consider the following statements. (i) The Laplace Transform of 11tet2 cos(et2) is well-defined for some values...
Consider the following statements.
(i) The Laplace Transform of
11tet2 cos(et2)
is well-defined for some values of s.
(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
continuous, or when it comes to studying some Volterra equations
and integro-differential equations.
(iii)...
just focus on A,B,D 1. Homogeneous ODE Find a general solution of the linear non-constant coefficient,...
just focus on A,B,D
1. Homogeneous ODE Find a general solution of the linear non-constant coefficient, homogeneous ODE for y(x) x3y'" – 3xy" + (6 – x2)xy' – (6 – x?)y = 0 as follows. a) You are given that yı(x) = x is a solution to the above homogeneous ODE. Confirm (by substitution) that this is the case. b) Apply reduction of order to find the remaining two solutions, then state the general solution. (Hint: The substitution y2(x) =...
2. You can use Dand write an operator instead of an equation in this question. (a) Find a constant coefficient linear homogeneous differential equation of lowest order that has n(x)-x , y2(z) = x2 ,...
2. You can use Dand write an operator instead of an equation in this question. (a) Find a constant coefficient linear homogeneous differential equation of lowest order that has n(x)-x , y2(z) = x2 , and y3(z) = eェamong its solutions. (b) Now find a different linear homogeneous differential equation of an order lower than the one in (a) that has the same y1,U2,U3 among its solutions. (c) Find a constant coefficient linear homogeneous differential equation of lowest order that...
IGNORE (i) (ii) The procedure of finding series solutions to a homogeneous linear second-order ODEs could...
IGNORE (i)
(ii) The procedure of finding series solutions to a homogeneous
linear second-order ODEs could be accurately described as the
“method of undetermined series coefficients”.
(iii) The underlying idea behind the method of undetermined
coefficients is a conjecture about the form of a particular
solution that is motivated by the right-hand side of the equation.
The method of undetermined coefficients is limited to second-order
linear ODEs with constant coefficients and the right-hand side of
the ODE cannot be an...
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...
8. (9 points) Suppose the characteristic equation of a certain twentieth order, linear, constant coefficient, homogeneous differential equation has roots: 2,0, a, 2+3i, ti, +4i, ti, 2, 3, a, 2+3i ,2,3,0, and -3. (where a is a real constant) Write the general solution to this differential equation. (Do not attempt to solve for the coefficients).
Problem #2: Consider the following statements. [6 marks) (1) The particular solution of the ODE)" - 6y' + 9y = 5e3x is given by yp = Cre3x where C is an undetermined constant. (ii) The procedure of finding series solutions to a homogeneous linear second-order ODEs could be accurately described as the "method of undetermined series coefficients". (iii) Most of the material in Lecture Notes from Week 3 to Week 5, inclusive, can be extended or generalized to higher-order ODES...
1. Second order ODE (25 points) a. Consider the following nonhomogeneous ODEs, find their homogeneous solution, and give the form (no need to determine coefficients) of nonhomogeneous solution. (12 points) i. 44'' + 3y = 4x sin ( *2) ii. J + 2 + 3 = eº cosh(22) b. Find the general solution of y" + 2Dy' + 2D'y = 5Dº cos(Dx) where D is a real constant with following steps i) Determine homogeneous solution, ii) Find nonhomogeneous solution with...
2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a vertical plane under the influence of gravity can be modelled with the second order non-homogeneous ODE 0"(t) + 270'(t) +w20(t) = f(t), (2) where wa = g/l. The second term in the equation represents the damping force (e.g. air resistance) for the given constant 7 > 0. The model can be used to approximate the motion of a magnetic pendulum bob being driven by...
(1) Use the "Schoolboy's Trick" to find a system of two ODE (first order homo geneous linear, with constant coefficients) in two "unknowns" r1,r2 which is "equivalent" to the second order homogeneous linear ODE with constant in the sense that establishes a bijection (one-to-one correspondence... in fact is is an "isomor- phism of vector spaces") from the set of solutions to the above ODE to the set of solutions to your system. Write your answer in the form Ar for...
Consider the following statements.
(i) The Laplace Transform of
11tet2 cos(et2)
is well-defined for some values of s.
(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
continuous, or when it comes to studying some Volterra equations
and integro-differential equations.
(iii)...
just focus on A,B,D
1. Homogeneous ODE Find a general solution of the linear non-constant coefficient, homogeneous ODE for y(x) x3y'" – 3xy" + (6 – x2)xy' – (6 – x?)y = 0 as follows. a) You are given that yı(x) = x is a solution to the above homogeneous ODE. Confirm (by substitution) that this is the case. b) Apply reduction of order to find the remaining two solutions, then state the general solution. (Hint: The substitution y2(x) =...
2. You can use Dand write an operator instead of an equation in this question. (a) Find a constant coefficient linear homogeneous differential equation of lowest order that has n(x)-x , y2(z) = x2 , and y3(z) = eェamong its solutions. (b) Now find a different linear homogeneous differential equation of an order lower than the one in (a) that has the same y1,U2,U3 among its solutions. (c) Find a constant coefficient linear homogeneous differential equation of lowest order that...
IGNORE (i)
(ii) The procedure of finding series solutions to a homogeneous
linear second-order ODEs could be accurately described as the
“method of undetermined series coefficients”.
(iii) The underlying idea behind the method of undetermined
coefficients is a conjecture about the form of a particular
solution that is motivated by the right-hand side of the equation.
The method of undetermined coefficients is limited to second-order
linear ODEs with constant coefficients and the right-hand side of
the ODE cannot be an...
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