cos 3x is a solution ==> 3i and -3i are roots of
the characteristic equation. So, we use synthetic division
to divide by 3i and -3i
Thus, the particular solution of the differential equation
y'' + y' = x + cos x + sin x is given by:
is a solution of the differential equation Ly = f(x)
is a solution of the Ly = g(x) (the same diff. eqn
with different non homogenous term), then y1 + y2
is a solution of Ly = f(x) + g(x).
Recall from homework problems 25 and 26 section 5.2
And the differential equation is
The roots of the characteristic polynomial are: 0,
0, 2i, -2i, 2i, -2i
Thus, the char. poly. is
Three vectors in R4
may not form a basis because the dimension of R4 is 4, which means that
any basis of R4
must consist of 4 vectors.
Since the determinant is zero ==> the vectors are
Thus, the particular solution is given by
Define f(x) (We must divide by x2 to put the equation
in standard form)
Compute the Wronskian of y1
We get two solutions from the complementary solution
But the cos 2x and sin 2x appear in the complementary
solution. So, we modify yp
by multiplying the cos and the sin terms by x to avoid
The form of the particular solution before modification.
Therefore, W is a subspace of R4
==> W is closed under scalar multiplication
Let c be an arbitrary constant.
since the first element of u+
is 3 time the third element and the second element
is 4 time the fourth element ==> u
is an element of W ==> W is closed under addition.
Let u and v be elements of W ==>
Since we have 3 vectors in R3, we can find out linear
dependence or independence by computing the determinant
of the augmented matrix [ v1
v2 v3 ]
The reduced row echelon form has a leading entry in
the last column ==> the system has no solution
==> w can not be written as a linear combination
of v1 and v2 ==> w is not a solution.
Compute the reduce row echelon form of Ag.
Augment the vectors and name the matrix Ag.
This is achieved by computing the reduced row echelon
form of the augmented matrix [v1
w]. Mathcad will be used to do the augmentation and
We need to solve the nonhomogeneous system
The dimension of the solution space is 2 since the basis
contains 2 vectors.
We can also use the Given ... Find command to solve
the system as shown below:
Check the computation of the reduced row echelon form
Thus, the basis vectors are