> #OK to post HW
> #Timothy Nasralla, 9/13/21, HW 2
> #Question 1, write a code to solve for the given sequence.
> #Similar to the first assignment I used a recursive function that proves to be effective in showing a(n) is equal to n^3.
;
> a := proc(n) option remember: 
> if n = 0 then
>  a = 0:
> elif n = 1 then
>  a = 1:
> elif n = 2 then
>  a = 8: 
> elif n = 3 then 
> a = 27: 
> else
>  4*a(n - 1) - 6*a(n - 2) + 4*a(n - 3) - a(n - 4):
> fi:
> end:
> 
;
> seq(a(n),n=0..8)
a = 0, a = 1, a = 8, a = 27, a = 64, a = 125, a = 216, a = 343, 



  a = 512




;
> #Question 2, solve the given ODE
> dsolve({diff(y(t),t)=(y(t))^3/(t+1),y(0)=1},y(t))
                                  1           

                 y(t) = ----------------------

                                         (1/2)

                        (1 - 2 ln(t + 1))     


;
> #Question 3, solve the given second order ODE with the given initial conditions
> dsolve({D(D(y))(t)-3*D(y)(t)+2*y(t)=0,y(0)=2,D(y)(0)=3},y(t))
                    y(t) = exp(2 t) + exp(t)


;
> #Question 4, find the eigenvalues and eigenvectors of the given 2x2 matrix
> with(LinearAlgebra);
[&x, Add, Adjoint, BackwardSubstitute, BandMatrix, Basis, 



  BezoutMatrix, BidiagonalForm, BilinearForm, CARE, 



  CharacteristicMatrix, CharacteristicPolynomial, Column, 



  ColumnDimension, ColumnOperation, ColumnSpace, CompanionMatrix, 



  CompressedSparseForm, ConditionNumber, ConstantMatrix, 



  ConstantVector, Copy, CreatePermutation, CrossProduct, DARE, 



  DeleteColumn, DeleteRow, Determinant, Diagonal, DiagonalMatrix, 



  Dimension, Dimensions, DotProduct, EigenConditionNumbers, 



  Eigenvalues, Eigenvectors, Equal, ForwardSubstitute, 



  FrobeniusForm, FromCompressedSparseForm, FromSplitForm, 



  GaussianElimination, GenerateEquations, GenerateMatrix, 



  Generic, GetResultDataType, GetResultShape, 



  GivensRotationMatrix, GramSchmidt, HankelMatrix, HermiteForm, 



  HermitianTranspose, HessenbergForm, HilbertMatrix, 



  HouseholderMatrix, IdentityMatrix, IntersectionBasis, 



  IsDefinite, IsOrthogonal, IsSimilar, IsUnitary, 



  JordanBlockMatrix, JordanForm, KroneckerProduct, LA_Main, 



  LUDecomposition, LeastSquares, LinearSolve, LyapunovSolve, Map, 



  Map2, MatrixAdd, MatrixExponential, MatrixFunction, 



  MatrixInverse, MatrixMatrixMultiply, MatrixNorm, MatrixPower, 



  MatrixScalarMultiply, MatrixVectorMultiply, MinimalPolynomial, 



  Minor, Modular, Multiply, NoUserValue, Norm, Normalize, 



  NullSpace, OuterProductMatrix, Permanent, Pivot, PopovForm, 



  ProjectionMatrix, QRDecomposition, RandomMatrix, RandomVector, 



  Rank, RationalCanonicalForm, ReducedRowEchelonForm, Row, 



  RowDimension, RowOperation, RowSpace, ScalarMatrix, 



  ScalarMultiply, ScalarVector, SchurForm, SingularValues, 



  SmithForm, SplitForm, StronglyConnectedBlocks, SubMatrix, 



  SubVector, SumBasis, SylvesterMatrix, SylvesterSolve, 



  ToeplitzMatrix, Trace, Transpose, TridiagonalForm, UnitVector, 



  VandermondeMatrix, VectorAdd, VectorAngle, 



  VectorMatrixMultiply, VectorNorm, VectorScalarMultiply, 



  ZeroMatrix, ZeroVector, Zip]




;
> B:=Matrix([[3,-4],[4,3]])
                           [3  &uminus0;4]

                      B := [             ]

                           [4      3     ]


;
> evalf(Eigenvectors(B))
 [3. + 4. ⅈ]  [ⅈ  &uminus0;ⅈ]

 [                    ], [                                   ]

 [3. - 4. ⅈ]  [     1.                1.          ]


;
> #The Eigenvectors function in evalf returns the eigenvalues and their respective eigenvectors from left to right.
;