3. (This option requires you to be able to differentiate exponential functions and hence will be...

 

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3. (This option requires you to be able to differentiate exponential functions and hence will be easier if you have completed SIT194 and SIT291). Eigenvectors can be used to solve systems of differential equations. For example, let y be a column vector whose entries are functions of t, i.e. y = The system of differential equations: or in matrix form, has the general solution dy dt i. Find the eigenvalues and eigenvectors of ayı (t) + by2(t) cy₁ (t) + dy2(t) y (0) = = a с y = k₁e¹₁¹x₁ + k₂e¹2x₂ where A₁, A2 are the eigenvalues and x₁, x2 are the eigenvectors of the matrix 61 ii. Choose k₁= 1 and k₂ = -1 and verify that the solution solves the system of differential equations. iii. Determine the values of k₁ and k₂ if we are given the initial conditions that d]y у iv. Use the same method to find the general solution to the following second order differential equation. Check that your solution seems correct by setting k₁= k₂ = 1 and subbing into the equation. y" + y - 6y=0 (Hint: you can set your two functions as y₁ (t) = y(t) and y2(t) = y'(t) and then express y"(t) in terms of y₁ and y2 - this should allow you to set up the system in the form above and obtain the general solution.) 10 marks 3. (This option requires you to be able to differentiate exponential functions and hence will be easier if you have completed SIT194 and SIT291). Eigenvectors can be used to solve systems of differential equations. For example, let y be a column vector whose entries are functions of t, i.e. y = The system of differential equations: or in matrix form, has the general solution dy dt i. Find the eigenvalues and eigenvectors of ayı (t) + by2(t) cy₁ (t) + dy2(t) y (0) = = a с y = k₁e¹₁¹x₁ + k₂e¹2x₂ where A₁, A2 are the eigenvalues and x₁, x2 are the eigenvectors of the matrix 61 ii. Choose k₁= 1 and k₂ = -1 and verify that the solution solves the system of differential equations. iii. Determine the values of k₁ and k₂ if we are given the initial conditions that d]y у iv. Use the same method to find the general solution to the following second order differential equation. Check that your solution seems correct by setting k₁= k₂ = 1 and subbing into the equation. y" + y - 6y=0 (Hint: you can set your two functions as y₁ (t) = y(t) and y2(t) = y'(t) and then express y"(t) in terms of y₁ and y2 - this should allow you to set up the system in the form above and obtain the general solution.) 10 marks

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