Question

What is the order of the following difference equation? Explain your answer\({y_{k + 3}} + {a_1}{y_{k + 2}} + {a_2}{y_{k + 1}} + {a_3}{y_k} = 0\).

Short answer

If \({a_3} = 0\) and \({a_2} \ne 0\), the order is 2.

If \({a_3} \ne 0\), the order is 3.

If \({a_3} = {a_2} = 0\) and \({a_1} \ne 0\), the order is 1.

If \({a_3} = {a_2} = {a_1} = 0\), the order is 0.

Step-by-step solution

Discuss the attributes of a difference equation

The given difference equation is \({y_{k + 3}} + {a_1}{y_{k + 2}} + {a_2}{y_{k + 1}} + {a_3}{y_k} = 0\). The order of a difference equation depends on the values of the coefficients \({a_1}\), \({a_2}\), and \({a_3}\).

Discuss the cases corresponding to the values of the coefficients

If \({a_3} = 0\) and \({a_2} \ne 0\), then the difference equation can be reduced to \({y_{k + 2}} + {a_1}{y_{k + !}} + {a_2}{y_k} = 0\), which implies that its order is 2.

If \({a_3} \ne 0\), then the difference equation cannot be reduced further, which implies that its order is 3.

If \({a_3} = {a_2} = 0\) and \({a_1} \ne 0\), then the difference equation can be reduced to \({y_{k + 1}} + {a_1}{y_k} = 0\), which implies that its order is 1.

If all the coefficients are zero, that is, \({a_3} = {a_2} = {a_1} = 0\), then the difference equation does not possess any signal value, which implies that its order is 0.

Draw a conclusion

If \({a_3} = 0\) and \({a_2} \ne 0\), the order is 2.

If \({a_3} \ne 0\), the order is 3.

If \({a_3} = {a_2} = 0\) and \({a_1} \ne 0\), the order is 1.

If \({a_3} = {a_2} = {a_1} = 0\), the order is 0.