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Chapter 4: Q31E (page 191)

Is the following difference equation of order 3? Explain.\({y_{k + 3}} + 5{y_{k + 2}} + 6{y_{k + 1}} = 0\).

Short Answer

Expert verified

No, the order of the given difference equation is 2.

Step by step solution

01

Replace \(k\)by \(k - 1\) and simplify the resulting equation

\(\begin{aligned} {y_{\left( {k - 1} \right) + 3}} + 5{y_{\left( {k - 1} \right) + 2}} + 6{y_{\left( {k - 1} \right) + 1}} &= 0\\{y_{k + 2}} + 5{y_{k + 1}} + 6{y_k} &= 0\end{aligned}\)

02

Compare the two equations for some value of \(k\)

For \(k = 2\), the given equation becomes \({y_5} + 5{y_4} + 6{y_3} = 0\). For \(k = 3\), the transformed equation becomes \({y_5} + 5{y_4} + 6{y_3} = 0\).

The second difference equation results in a transformed one for each successive value of \(k\). It means that both equations are true for all the natural values assumed for \(k\).

03

Draw a conclusion

As the difference equation \({y_{k + 2}} + 5{y_{k + 1}} + 6{y_k} = 0\) is true for all \(k\), the order of the difference equation is 2.

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Most popular questions from this chapter

If the null space of a\({\bf{5}} \times {\bf{6}}\)matrix A is 4-dimensional, what is the dimension of the row space of A?

Prove theorem 3 as follows: Given an \(m \times n\) matrix A, an element in \({\mathop{\rm Col}\nolimits} A\) has the form \(Ax\) for some x in \({\mathbb{R}^n}\). Let \(Ax\) and \(A{\mathop{\rm w}\nolimits} \) represent any two vectors in \({\mathop{\rm Col}\nolimits} A\).

  1. Explain why the zero vector is in \({\mathop{\rm Col}\nolimits} A\).
  2. Show that the vector \(A{\mathop{\rm x}\nolimits} + A{\mathop{\rm w}\nolimits} \) is in \({\mathop{\rm Col}\nolimits} A\).
  3. Given a scalar \(c\), show that \(c\left( {A{\mathop{\rm x}\nolimits} } \right)\) is in \({\mathop{\rm Col}\nolimits} A\).

Let S be a maximal linearly independent subset of a vector space V. In other words, S has the property that if a vector not in S is adjoined to S, the new set will no longer be linearly independent. Prove that S must be a basis of V. [Hint: What if S were linearly independent but not a basis of V?]

If a\({\bf{6}} \times {\bf{3}}\)matrix A has a rank 3, find dim Nul A, dim Row A, and rank\({A^T}\).

[M] Let \(A = \left[ {\begin{array}{*{20}{c}}7&{ - 9}&{ - 4}&5&3&{ - 3}&{ - 7}\\{ - 4}&6&7&{ - 2}&{ - 6}&{ - 5}&5\\5&{ - 7}&{ - 6}&5&{ - 6}&2&8\\{ - 3}&5&8&{ - 1}&{ - 7}&{ - 4}&8\\6&{ - 8}&{ - 5}&4&4&9&3\end{array}} \right]\).

  1. Construct matrices \(C\) and \(N\) whose columns are bases for \({\mathop{\rm Col}\nolimits} A\) and \({\mathop{\rm Nul}\nolimits} A\), respectively, and construct a matrix \(R\) whose rows form a basis for Row\(A\).
  2. Construct a matrix \(M\) whose columns form a basis for \({\mathop{\rm Nul}\nolimits} {A^T}\), form the matrices \(S = \left[ {\begin{array}{*{20}{c}}{{R^T}}&N\end{array}} \right]\) and \(T = \left[ {\begin{array}{*{20}{c}}C&M\end{array}} \right]\), and explain why \(S\) and \(T\) should be square. Verify that both \(S\) and \(T\) are invertible.
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