Using Euler's method (the method of eigenvalues and eigenvectors, the case of different complex eigenvalues) solve the sytem of differential equations y ′ → = A y → or solve the initial value problem y ′ → = A y → , y → (0) = y → 0 , if [ − 7 1 − 2 − 5 ] My attempt: (but unfortunately it is not Euler's method, i don't understand this method):So i have started from forming a new matrix by subtracting λ from the diagonal entries of the given matrix: [ − λ − 7 1 − 2 − λ − 5 ] The determinant of the obtained matrix is λ 2 + 12 λ + 37Solve the equation λ 2 + 12 λ + 37 = 0.The roots are λ 1 = − 6 − i , λ 2 = − 6 + i, these are the eigenvalues.Next, i have found the eigenvectors.1. λ = − 6 − i [ − λ − 7 1 − 2 − λ − 5 ] = [ − 1 + i 1 − 2 i + 1 ] The null space of this matrix is [ 1 / 2 + i / 2 1 ] This is the eigenvector.2. λ = − 6 + i [ − λ − 7 1 − 2 − λ − 5 ] = [ − 1 + i 1 − 2 i − 1 ] The null space of this matrix is [ 1 / 2 − i / 2 1 ] and this is the eigenvector.And i dont know what i should do next to calculate the fundamental equations (formula) or any other solution is of the following form

Question

Using Euler&#039;s method (the method of eigenvalues and eigenvectors, the case of different complex eigenvalues) solve the sytem of differential equations             y      ′        →   = A      y    →   or solve the initial value problem             y      ′        →  = A      y    →  ,       y    →  (0) =             y      →        0  , if      [                            −          7                          1                                      −          2                          −          5                      ]  My attempt: (but unfortunately it is not Euler&#039;s method, i don&#039;t understand this method):So i have started from forming a new matrix by subtracting       λ   from the diagonal entries of the given matrix:      [                            −                      λ                    −          7                          1                                      −          2                          −                      λ                    −          5                      ]  The determinant of the obtained matrix is             λ        2    +  12      λ    +  37Solve the equation             λ        2    +  12      λ    +  37  =  0.The roots are       λ    1  =  −  6  −  i  ,      λ    2  =  −  6  +  i, these are the eigenvalues.Next, i have found the eigenvectors.1.       λ    =  −  6  −  i      [                            −                      λ                    −          7                          1                                      −          2                          −                      λ                    −          5                      ]    =      [                            −          1          +          i                          1                                      −          2                          i          +          1                      ]  The null space of this matrix is       [                            1                      /                    2          +          i                      /                    2                                      1                      ]  This is the eigenvector.2.       λ    =  −  6  +  i      [                            −                      λ                    −          7                          1                                      −          2                          −                      λ                    −          5                      ]    =      [                            −          1          +          i                          1                                      −          2                          i          −          1                      ]  The null space of this matrix is       [                            1                      /                    2          −          i                      /                    2                                      1                      ]  and this is the eigenvector.And i dont know what i should do next to calculate the fundamental equations (formula) or any other solution is of the following form

Jazmin Cameron · Accepted Answer

Hint: For linear homogeneous differential equations we have the following observation. If   z  ∈      C   is a solution to the differential, then we know that the real and imaginary part for   z are also solutions to the differential equation.Hence, calculate the complex solution first. Then determine the real and imaginary part of your base solutions you should be able to extract two independent solutions from this step.

Using Euler's method (the method of eigenvalues and eigenvectors, the case of different complex eigenvalues) solve the sytem of differential equations vec y' = A vec y or solve the initial value problem vec y'= A vec y, vec y(0) = vec y_0, if [−7−21−5]

Open question

Answer & Explanation

New Questions in Integral Calculus