For what value of x must ABCD be a parallelogram?
2 answers:
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Rearrange the ODE as
Take
, so that 
.
Supposing that
, we have 
, from which it follows that
So we can write the ODE as
which is linear in
. Multiplying both sides by 
, we have
![\dfrac{\mathrm d}{\mathrm dx}\bigg[e^{x^2}u\bigg]=x^3e^{x^2}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cbigg%5Be%5E%7Bx%5E2%7Du%5Cbigg%5D%3Dx%5E3e%5E%7Bx%5E2%7D)
Integrate both sides with respect to
:
![\displaystyle\int\frac{\mathrm d}{\mathrm dx}\bigg[e^{x^2}u\bigg]\,\mathrm dx=\int x^3e^{x^2}\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cbigg%5Be%5E%7Bx%5E2%7Du%5Cbigg%5D%5C%2C%5Cmathrm%20dx%3D%5Cint%20x%5E3e%5E%7Bx%5E2%7D%5C%2C%5Cmathrm%20dx)
Substitute
, so that 
. Then
Integrate the right hand side by parts using
You should end up with
and provided that we restrict, we can write
Take
Supposing that
So we can write the ODE as
which is linear in
Integrate both sides with respect to
Substitute
Integrate the right hand side by parts using
You should end up with
and provided that we restrict
The row echelon form of the matrix is presented as follows;
The row echelon form of a matrix has the rows consisting entirely of zeros at the bottom, and the first entry of a row that is not entirely zero is a one.
The given matrix is presented as follows;
The conditions of a matrix in the row echelon form are as follows;
- There are row having nonzero entries above the zero rows.
- The first nonzero entry in a nonzero row is a one.
- The location of the leading one in a nonzero row is to the left of the leading one in the next lower rows.
Dividing Row 1 by -3 gives:
Multiplying Row 1 by 2 and subtracting the result from Row 2 gives;
Subtracting Row 1 from Row 3 gives;
Adding Row 2 to Row 3 gives;
Dividing Row 2 by -2, and Row 3 by 18 gives;
The above matrix is in the row echelon form
Learn more about the row echelon form here:
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