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I need help with these! 3. (1 point) a) Compute the general solution of the differential equation y"5 12y" 0 b) Determine the test function Y (t) with the fewest terms to be used to obtain...
(4) (12 points)For the differential equation: Compute the recursion formula for the coefficients of the power series solution centered at o 0 and use it to compute the first three nonzero terms of...
(4) (12 points)For the differential equation: Compute the recursion formula for the coefficients of the power series solution centered at o 0 and use it to compute the first three nonzero terms of the solution with y(0) 12, y'(0)0. (5) (12 points)For the equation y" - 5ty -7y 0 (t>0), (t)t is a solution (a) Use the method of Reduction of Order to obtain a second, independent solution. (b) Solve the equation directly, using that it is an Euler Equation....
Consider the ordinary differential equation: t2y" + 3ty' +y = 0. 1 (3 points) e) Use...
Consider the ordinary differential equation: t2y" + 3ty' +y = 0. 1 (3 points) e) Use Abel's formula to find the Wronskian of any two solutions of this equation and W[y1,y2](t). What do you observe? compare it to = t1 and y2(t) = t-1 nt represent a fundamental set of solu f) (2 points) Determine if y1 (t) tions (2 points) Find the general solution of t2y" +3ty' +y = 0. g) Solve the initial value problem t2y" + 3ty/...
Find the general solution of the differential equation nts y" – 7y' +12y = 2te3+ +1...
Find the general solution of the differential equation nts y" – 7y' +12y = 2te3+ +1 - t y(t) = 20e3+ + boe4 – 2te3 – 12e31 - í y(t) = ape?+ boet! – 2te3 – 12e3! + y(t) = do Cost + bo sint + fr sint - t? cost – cos(21) y(t) = ao + boe?' – št-5te3+ + 372e34 y(t) = (ao + ajt)e3! + 312e3!
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the s...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
(1 point) Find the indicated coefficients of the power series solution about x-0 of the differential equation (x2-x+1y"-y-3y = 0, y(0) = 0, y(o) =-8 x2+ 4 (1 point) Find the indicated coeffi...
(1 point) Find the indicated coefficients of the power series solution about x-0 of the differential equation (x2-x+1y"-y-3y = 0, y(0) = 0, y(o) =-8 x2+ 4
(1 point) Find the indicated coefficients of the power series solution about x-0 of the differential equation (x2-x+1y"-y-3y = 0, y(0) = 0, y(o) =-8 x2+ 4
Find a general solution to the given differential equation. 21y'' + 8y' - 5y = 0 A general solution is y(t) = _______ .
1.Find a general solution to the given differential equation. 21y'' + 8y' - 5y = 0 A general solution is y(t) = _______ .2.Solve the given initial value problem. y'' + 3y' = 0; y(0) = 12, y'(0)= - 27 The solution is y(t) = _______ 3.Find three linearly independent solutions of the given third-order differential equation and write a general solution as an arbitrary linear combination of them z"'+z"-21z'-45z = 0 A general solution is z(t) = _______
1. Compute the Wronskian for the following functions. Then use the Wronskian to determine whether the...
1. Compute the Wronskian for the following functions. Then use the Wronskian to determine whether the functions are linearly independant or linearly dependant. a) {(tan2x - sec2 x),3 (b) le,e,e) 2. Use variation of parameters to find a general solution to 2y" -4ry 6y3 1 given that y 2 and y2- 3 are linearly independant solutions of the associated homogeneous equation. (Hint: be careful the equations are in the right form.) Find a particular solution for each of the following...
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and ide...
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and identify the matrix A. Obtain a general solution to the matrix differential equation in terms of eigenvectors and eigenvalues of A. Justify your answer (b) Classify possible types and stability of the equilibrium with dependence on the eigenvalues of A. (Note: You are not asked to compute the...
m FE18 Consider the second-ord er inhomogeneous differential equation -10cos(t/3)-10sin(t/3), y"-y ith the initial conditions y(0)...
m FE18 Consider the second-ord er inhomogeneous differential equation -10cos(t/3)-10sin(t/3), y"-y ith the initial conditions y(0) = 1/2 and y,(0) =-1/2 ) (2 points) Obtain the general solution, yelt), of the corresponding homog ) (6 points) Using the method of undetermined coefficients, obtain a soluti geneous equation, Y (t), and write down the general solution of equation 5
3. Consider the differential equation ty" - (t+1)y + y = t?e?', t>0. (a) Find a...
3. Consider the differential equation ty" - (t+1)y + y = t?e?', t>0. (a) Find a value ofr for which y = et is a solution to the corresponding homogeneous differential equation. (b) Use Reduction of Order to find a second, linearly independent, solution to the correspond- ing homogeneous differential equation. (c) Use Variation of Parameters to find a particular solution to the nonhomogeneous differ- ential equation and then give the general solution to the differential equation.
(4) (12 points)For the differential equation: Compute the recursion formula for the coefficients of the power series solution centered at o 0 and use it to compute the first three nonzero terms of the solution with y(0) 12, y'(0)0. (5) (12 points)For the equation y" - 5ty -7y 0 (t>0), (t)t is a solution (a) Use the method of Reduction of Order to obtain a second, independent solution. (b) Solve the equation directly, using that it is an Euler Equation....
Consider the ordinary differential equation: t2y" + 3ty' +y = 0. 1 (3 points) e) Use Abel's formula to find the Wronskian of any two solutions of this equation and W[y1,y2](t). What do you observe? compare it to = t1 and y2(t) = t-1 nt represent a fundamental set of solu f) (2 points) Determine if y1 (t) tions (2 points) Find the general solution of t2y" +3ty' +y = 0. g) Solve the initial value problem t2y" + 3ty/...
Find the general solution of the differential equation nts y" – 7y' +12y = 2te3+ +1 - t y(t) = 20e3+ + boe4 – 2te3 – 12e31 - í y(t) = ape?+ boet! – 2te3 – 12e3! + y(t) = do Cost + bo sint + fr sint - t? cost – cos(21) y(t) = ao + boe?' – št-5te3+ + 372e34 y(t) = (ao + ajt)e3! + 312e3!
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
(1 point) Find the indicated coefficients of the power series solution about x-0 of the differential equation (x2-x+1y"-y-3y = 0, y(0) = 0, y(o) =-8 x2+ 4
(1 point) Find the indicated coefficients of the power series solution about x-0 of the differential equation (x2-x+1y"-y-3y = 0, y(0) = 0, y(o) =-8 x2+ 4
1. Compute the Wronskian for the following functions. Then use the Wronskian to determine whether the functions are linearly independant or linearly dependant. a) {(tan2x - sec2 x),3 (b) le,e,e) 2. Use variation of parameters to find a general solution to 2y" -4ry 6y3 1 given that y 2 and y2- 3 are linearly independant solutions of the associated homogeneous equation. (Hint: be careful the equations are in the right form.) Find a particular solution for each of the following...
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and identify the matrix A. Obtain a general solution to the matrix differential equation in terms of eigenvectors and eigenvalues of A. Justify your answer (b) Classify possible types and stability of the equilibrium with dependence on the eigenvalues of A. (Note: You are not asked to compute the...
m FE18 Consider the second-ord er inhomogeneous differential equation -10cos(t/3)-10sin(t/3), y"-y ith the initial conditions y(0) = 1/2 and y,(0) =-1/2 ) (2 points) Obtain the general solution, yelt), of the corresponding homog ) (6 points) Using the method of undetermined coefficients, obtain a soluti geneous equation, Y (t), and write down the general solution of equation 5
3. Consider the differential equation ty" - (t+1)y + y = t?e?', t>0. (a) Find a value ofr for which y = et is a solution to the corresponding homogeneous differential equation. (b) Use Reduction of Order to find a second, linearly independent, solution to the correspond- ing homogeneous differential equation. (c) Use Variation of Parameters to find a particular solution to the nonhomogeneous differ- ential equation and then give the general solution to the differential equation.
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