Find the general solution of the differential equation
$$3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = - 2 x + 13 .$$
Find the particular solution for which \(y = - \frac { 7 } { 2 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).
Write down the function to which \(y\) approximates when \(x\) is large and positive.
\(6 Q\) is a multiplicative group of order 12.
Two elements of \(Q\) are \(a\) and \(r\). It is given that \(r\) has order 6 and that \(a ^ { 2 } = r ^ { 3 }\). Find the orders of the elements \(a , a ^ { 2 } , a ^ { 3 }\) and \(r ^ { 2 }\).
The table below shows the number of elements of \(Q\) with each possible order.
Order of element
1
2
3
4
6
Number of elements
1
1
2
6
2
\(G\) and \(H\) are the non-cyclic groups of order 4 and 6 respectively.
Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups \(G\) and \(H\). Hence explain why there are no non-cyclic proper subgroups of \(Q\).