problem 1 , find the general solution 
differential equation
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problem 1 , find the general solution
differential equation
9. 2't) = = (Å -1) =(e)...
find the general solution differential equation 13. 2' (t) = | r(t) (3-1( (1 21) (2)...
find the general solution
differential equation
13. 2' (t) = | r(t) (3-1( (1 21) (2) 14. :'(t) = -5 15. :'(t) = 10 0 2 1 -2(t) 32 -1 16. :'(t) = '-3 0 2 1 -1 0r(t) -2 -1 Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0 for some to El, then f and g are linearly independent for all t E I. If f(t) and...
9. :'(t) = = (Å -1) =(e) 10. aº(t) = (1 -1) =(e) 11. 2°(t) =...
9. :'(t) = = (Å -1) =(e) 10. aº(t) = (1 -1) =(e) 11. 2°(t) = ({ ->) (0) 12.740 = C +1) Rio (t) 13. :'(t) = 5 14. I'(t) = (45 15 3)=(0) 0 15. r'(t) = 1 0 2 1 3 2 -2 r(t) -1 16. :'(t) = -3 0 2 1 -1 0 2(t) -2 -1 Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0...
find the general solution for 6,7,8 (differential equation) 6. L'(t) = 1 1 -1 r(t) -3...
find the general solution for 6,7,8
(differential equation)
6. L'(t) = 1 1 -1 r(t) -3 -8 -5 3 2 4 7. :'(t) = 2 0 2 r(t) 4 2 3 1 8. r'(t) = 3 2 -1 2 1 4 -1 (t) Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0 for some to El, then f and g are linearly independent for all t E I. If...
find the general solution for 14,15,16 differential equation 14. z't) 1 -5 21) =(0) 15. z'(t)...
find the general solution for 14,15,16
differential equation
14. z't) 1 -5 21) =(0) 15. z'(t) = 10 0 2 1 -2 (t) 3 2 -1 16. x'(t) = -3 0 2 1 -1 0 (t) -2 -1 0 Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0 for some to El, then f and g are linearly independent for all t E I. If f(t) and g(t) are...
Problem 1 Find the general solution: 9. 7(t) = 3 4 ar(t) ) (4) () 10....
Problem 1 Find the general solution:
9. 7(t) = 3 4 ar(t) ) (4) () 10. I'(t) 11. It) = 2. | z (t) 12. (t) = ( ) (1) Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0 for some to El, then f and g are linearly independent for allt el. If f(t) and g(t) are linearly dependent on I, then W(f:9)(t) = 0 for all tel....
Problem 1 Find the general solution: 13. :'(t) = (6 - 3) =(0) (45 21) =()...
Problem 1 Find the general solution:
13. :'(t) = (6 - 3) =(0) (45 21) =() 14. I'(t) = 15. r'(t) = 1 0 0 2 1 -2 (t) 3 2 -1 16. :'(t) = -3 0 2 1 -1 0 (t) -2 -1 Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0 for some to El, then f and g are linearly independent for allt el. If f(t)...
Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If...
Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0 for some to E I, then f and g are linearly independent for all te I. • If f(t) and g(t) are linearly dependent on I, then W (8,9)(t) = 0 for allt € 1. Note: This does NOT say that "If W(8,9)(x) = 0, then f(x) and g(2) are linearly dependent. Problem 2 Determine if the following functions are...
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) = _______
Problem 2 Determine if the following functions are linearly independent or linearly dependent. If you believe...
Problem 2 Determine if the following functions are linearly independent or linearly dependent. If you believe that they are linearly dependent (i.e. W(5,9) (+) = 0, for all t in some interval) find a dependence relation. 1. f(t) = cost, g(t) = sint 2. f(t) = 61, g(t) = 64+2 3. f(t) = 9 cos 2t, g(t) = 2 cos? t - 2 sinat 4. f(t) = 2t>, g(t) = 14
Please prove this solution and explain why y2 can be taken as (x^2)(y1) Problem 2. Find the general solution of the equation Note that one of two linearly independent solutions is yi(r) -e. Solution....
Please prove this solution and explain why y2 can be taken as
(x^2)(y1)
Problem 2. Find the general solution of the equation Note that one of two linearly independent solutions is yi(r) -e. Solution. Using Abel's formula, we get the following relations for the Wronskian dW pi dW 2r1 On the other hand, Comparing these two expression for W(x), we can take y2 :- r2yı. Correspondingly, the general solution is
Problem 2. Find the general solution of the equation Note...
find the general solution
differential equation
13. 2' (t) = | r(t) (3-1( (1 21) (2) 14. :'(t) = -5 15. :'(t) = 10 0 2 1 -2(t) 32 -1 16. :'(t) = '-3 0 2 1 -1 0r(t) -2 -1 Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0 for some to El, then f and g are linearly independent for all t E I. If f(t) and...
9. :'(t) = = (Å -1) =(e) 10. aº(t) = (1 -1) =(e) 11. 2°(t) = ({ ->) (0) 12.740 = C +1) Rio (t) 13. :'(t) = 5 14. I'(t) = (45 15 3)=(0) 0 15. r'(t) = 1 0 2 1 3 2 -2 r(t) -1 16. :'(t) = -3 0 2 1 -1 0 2(t) -2 -1 Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0...
find the general solution for 6,7,8
(differential equation)
6. L'(t) = 1 1 -1 r(t) -3 -8 -5 3 2 4 7. :'(t) = 2 0 2 r(t) 4 2 3 1 8. r'(t) = 3 2 -1 2 1 4 -1 (t) Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0 for some to El, then f and g are linearly independent for all t E I. If...
find the general solution for 14,15,16
differential equation
14. z't) 1 -5 21) =(0) 15. z'(t) = 10 0 2 1 -2 (t) 3 2 -1 16. x'(t) = -3 0 2 1 -1 0 (t) -2 -1 0 Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0 for some to El, then f and g are linearly independent for all t E I. If f(t) and g(t) are...
Problem 1 Find the general solution:
9. 7(t) = 3 4 ar(t) ) (4) () 10. I'(t) 11. It) = 2. | z (t) 12. (t) = ( ) (1) Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0 for some to El, then f and g are linearly independent for allt el. If f(t) and g(t) are linearly dependent on I, then W(f:9)(t) = 0 for all tel....
Problem 1 Find the general solution:
13. :'(t) = (6 - 3) =(0) (45 21) =() 14. I'(t) = 15. r'(t) = 1 0 0 2 1 -2 (t) 3 2 -1 16. :'(t) = -3 0 2 1 -1 0 (t) -2 -1 Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0 for some to El, then f and g are linearly independent for allt el. If f(t)...
Recall: Given two functions f(t) and g(t), which are differentiable on an interval I, • If the Wronskian W(8,9)(to) #0 for some to E I, then f and g are linearly independent for all te I. • If f(t) and g(t) are linearly dependent on I, then W (8,9)(t) = 0 for allt € 1. Note: This does NOT say that "If W(8,9)(x) = 0, then f(x) and g(2) are linearly dependent. Problem 2 Determine if the following functions are...
Problem 2 Determine if the following functions are linearly independent or linearly dependent. If you believe that they are linearly dependent (i.e. W(5,9) (+) = 0, for all t in some interval) find a dependence relation. 1. f(t) = cost, g(t) = sint 2. f(t) = 61, g(t) = 64+2 3. f(t) = 9 cos 2t, g(t) = 2 cos? t - 2 sinat 4. f(t) = 2t>, g(t) = 14
Please prove this solution and explain why y2 can be taken as
(x^2)(y1)
Problem 2. Find the general solution of the equation Note that one of two linearly independent solutions is yi(r) -e. Solution. Using Abel's formula, we get the following relations for the Wronskian dW pi dW 2r1 On the other hand, Comparing these two expression for W(x), we can take y2 :- r2yı. Correspondingly, the general solution is
Problem 2. Find the general solution of the equation Note...
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