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6. (20 points) Consider the following non-homogeneous set of coupled, ordinary differential equat...
Differential Equations for Engineers II Page 2 of 6 2. Consider the nonhomogeneous ordinary differential equation...
Differential Equations for Engineers II Page 2 of 6 2. Consider the nonhomogeneous ordinary differential equation XY" + 2(x – B)y' + (x – 2B)y = e-1, x > 0, (2) 5 marks where ß > 0 is a given constant. (a) A solution of the associated homogeneous equation is yı = e-*. Use the formula for the method of reduction of order, as described in the lecture notes / record- ings, to find a second solution, y2, of the...
Consider the following non-homogeneous system of differential equations. a. Write the system in matrix form. b....
Consider the following non-homogeneous system of differential
equations.
a. Write the system in matrix form.
b. Find the homogeneous solution.
c. Find the particular solution.
d. Write down the general solution.
We were unable to transcribe this imageWe were unable to transcribe this image
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...
(1 point) In general for a non-homogeneous problem " ()y r)y-f(x) assume that yi, ye is...
(1 point) In general for a non-homogeneous problem " ()y r)y-f(x) assume that yi, ye is a fundamental set of solutions for the homogeneous problem y"+p(r)y' +(xy-0. Then the formula for the particular solution using the method of variation of parameters is are where W(z) is the Wronskian given by the determinant where ufe) and u ,-1-nent), d dz. NOTE When evaluating these indefinite integrals we take the arbitrary constant of integration to be zero. So we have- Wed and...
As a specific example we consider the non-homogeneous problem y"+9y sec (3) (1) The general solution...
As a specific example we consider the non-homogeneous problem y"+9y sec (3) (1) The general solution of the homogeneous problem (called the complementary solution, sab2) is gliven in terms of a pair of linearly independent solutions, y1W Here α and b are arbitrary constants. Find a fundamental set for y"+9y -0 and enter your results as a comma separated list BEWARE Ntice that the above set does not require you to decide which function is to be called y or...
(1 point In general for a non-homogeneous problem y' + p(x) +(z) = f() assume that...
(1 point In general for a non-homogeneous problem y' + p(x) +(z) = f() assume that y. is a fundamental set of solutions for the homogeneous problemy" p(x) + (2) 0. Then the formula for the particular solution using the method of variation of parameters is where (z)/ and ()/() where W() is the Wronskian given by the determinant W (2) (2) W2) 31(2)/(2) dr. NOTE When evaluating these indefinite integrals we take the W(2) So we have the de...
4. Find the general solution to each of the following non- homogeneous second order ODES. d²y...
4. Find the general solution to each of the following non- homogeneous second order ODES. d²y dy -2+ y = -x + 3 dx dx2 Hint: Use the method of undetermined coefficients in finding the particular solutio day b) dx2 + y = secx Hint: Use variation of parameters for finding the particular solution. > The following problem is for bonus points. -- Solve the following ODE: dy + 5y = 10e-5x dx
As a specific example we consider the non-homogeneous problem y" +9y' + 18y = 9 sin(32)...
As a specific example we consider the non-homogeneous problem y" +9y' + 18y = 9 sin(32) (1) The general solution of the homogeneous problem (called the complementary solution, yc = ayı + by2 ) is given in terms of a pair of linearly independent solutions, 41, 42. Here a and b are arbitrary constants. Find a fundamental set for y" +9y' + 18y = 0 and enter your results as a comma separated list e^(-3x), e^(-x) BEWARE Notice that the...
Problem Set A Problem 6. (20%) A ordinary differential equation for a mass-damper-spring system is following....
Problem Set A Problem 6. (20%) A ordinary differential equation for a mass-damper-spring system is following. The mass m 1, damping coetfic e initial position y(o) O, and the initial velocity i constant k 3 and force 10, all are in appropriate units. Th 1, spring zero, within the time range of O to 20 unit of time, use Matlab find the solution of function y(t)? Hint: you need to convert the 2nd order ODE into two 1st order ODEs....
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...
Differential Equations for Engineers II Page 2 of 6 2. Consider the nonhomogeneous ordinary differential equation XY" + 2(x – B)y' + (x – 2B)y = e-1, x > 0, (2) 5 marks where ß > 0 is a given constant. (a) A solution of the associated homogeneous equation is yı = e-*. Use the formula for the method of reduction of order, as described in the lecture notes / record- ings, to find a second solution, y2, of the...
Consider the following non-homogeneous system of differential
equations.
a. Write the system in matrix form.
b. Find the homogeneous solution.
c. Find the particular solution.
d. Write down the general solution.
We were unable to transcribe this imageWe were unable to transcribe this image
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...
(1 point) In general for a non-homogeneous problem " ()y r)y-f(x) assume that yi, ye is a fundamental set of solutions for the homogeneous problem y"+p(r)y' +(xy-0. Then the formula for the particular solution using the method of variation of parameters is are where W(z) is the Wronskian given by the determinant where ufe) and u ,-1-nent), d dz. NOTE When evaluating these indefinite integrals we take the arbitrary constant of integration to be zero. So we have- Wed and...
As a specific example we consider the non-homogeneous problem y"+9y sec (3) (1) The general solution of the homogeneous problem (called the complementary solution, sab2) is gliven in terms of a pair of linearly independent solutions, y1W Here α and b are arbitrary constants. Find a fundamental set for y"+9y -0 and enter your results as a comma separated list BEWARE Ntice that the above set does not require you to decide which function is to be called y or...
(1 point In general for a non-homogeneous problem y' + p(x) +(z) = f() assume that y. is a fundamental set of solutions for the homogeneous problemy" p(x) + (2) 0. Then the formula for the particular solution using the method of variation of parameters is where (z)/ and ()/() where W() is the Wronskian given by the determinant W (2) (2) W2) 31(2)/(2) dr. NOTE When evaluating these indefinite integrals we take the W(2) So we have the de...
4. Find the general solution to each of the following non- homogeneous second order ODES. d²y dy -2+ y = -x + 3 dx dx2 Hint: Use the method of undetermined coefficients in finding the particular solutio day b) dx2 + y = secx Hint: Use variation of parameters for finding the particular solution. > The following problem is for bonus points. -- Solve the following ODE: dy + 5y = 10e-5x dx
As a specific example we consider the non-homogeneous problem y" +9y' + 18y = 9 sin(32) (1) The general solution of the homogeneous problem (called the complementary solution, yc = ayı + by2 ) is given in terms of a pair of linearly independent solutions, 41, 42. Here a and b are arbitrary constants. Find a fundamental set for y" +9y' + 18y = 0 and enter your results as a comma separated list e^(-3x), e^(-x) BEWARE Notice that the...
Problem Set A Problem 6. (20%) A ordinary differential equation for a mass-damper-spring system is following. The mass m 1, damping coetfic e initial position y(o) O, and the initial velocity i constant k 3 and force 10, all are in appropriate units. Th 1, spring zero, within the time range of O to 20 unit of time, use Matlab find the solution of function y(t)? Hint: you need to convert the 2nd order ODE into two 1st order ODEs....
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...
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