Find the solution of the differential equation
\(\frac{d y}{d x}-\frac{2 x+6}{x^{2}+6 x+5} y=4,\)
given that \(y=0\) when \(x=0\). Give your answer in an exact form.

Find the solution of the differential equation
\(\frac{\mathrm{d} y}{\mathrm{~d} x}-\frac{x+5}{x^{2}+10 x+61} y=1,\)
given that \(y=0\) when \(x=3\). Give your answer in an exact form.

(a) Show that an appropriate integrating factor for
\(\sqrt{x^{2}+16} \frac{\mathrm{~d} y}{\mathrm{~d} x}+y=x \sqrt{x^{2}+16}\)
is \(\frac{1}{4} x+\frac{1}{4} \sqrt{x^{2}+16}\).
(b) Hence find the solution of the differential equation
\(\sqrt{x^{2}+16} \frac{\mathrm{~d} y}{\mathrm{~d} x}+y=x \sqrt{x^{2}+16}\)
for which \(y=6\) when \(x=3\).

Find the solution of the differential equation
\(\frac{\mathrm{d} y}{\mathrm{~d} x}+3 y=\sin x\)
for which \(y=1\) when \(x=0\). Give your answer in the form \(y=\mathrm{f}(x)\).

Find the solution of the differential equation
\(x(x+7) \frac{\mathrm{d} y}{\mathrm{~d} x}+7 y=x\)
for which \(y=7\) when \(x=1\). Give your answer in the form \(y=\mathrm{f}(x)\).

Find the solution of the differential equation
\(\left(4t^2-1\right)\frac{\mathrm{d}x}{\mathrm{d}t}+4x=4t^2-1\)

for which \(x=3\) when \(t=1\). Give your answer in the form \(x=f(t)\).

Find the solution of the differential equation
\(\sin \theta \frac{\mathrm{d} y}{\mathrm{~d} \theta}+y=\tan \frac{1}{2} \theta,\)
where \(0\lt \theta\lt \pi\), given that \(y=1\) when \(\theta=\frac{1}{2} \pi\). Give your answer in the form \(y=\mathrm{f}(\theta)\).

(a) Show that an appropriate integrating factor for
\(\sqrt{x^{2}-1} \frac{\mathrm{~d} y}{\mathrm{~d} x}+y=x^{2}-x \sqrt{x^{2}-1}\)
is \(x+\sqrt{x^{2}-1}\).

(b) Hence find the solution of the differential equation
\(\sqrt{x^{2}-1} \frac{\mathrm{~d} y}{\mathrm{~d} x}+y=x^{2}-x \sqrt{x^{2}-1}\)
for which \(y=1\) when \(x=\frac{5}{4}\). Give your answer in the form \(y=\mathrm{f}(x)\).

Find the solution of the differential equation
\[\frac{\mathrm{d} y}{\mathrm{~d} x}+5 y=\mathrm{e}^{-7 x}\]
for which \(y=0\) when \(x=0\). Give your answer in the form \(y=\mathrm{f}(x)\).

(a) Show that an appropriate integrating factor for
\[\left(x^{2}+1\right) \frac{\mathrm{d} y}{\mathrm{~d} x}+y \sqrt{x^{2}+1}=x^{2}-x \sqrt{x^{2}+1}\]
is \(x+\sqrt{x^{2}+1}\).
(b) Hence find the solution of the differential equation
\[\left(x^{2}+1\right) \frac{\mathrm{d} y}{\mathrm{~d} x}+y \sqrt{x^{2}+1}=x^{2}-x \sqrt{x^{2}+1}\]
for which \(y=\ln 2\) when \(x=0\). Give your answer in the form \(y=\mathrm{f}(x)\).

(a) Starting from the definitions of sinh and cosh in terms of exponentials, prove that
\[2 \sinh ^{2} x=\cosh 2 x-1\]
(b) Find the solution to the differential equation
\[\frac{\mathrm{d} y}{\mathrm{~d} x}+y \operatorname{coth} x=4 \sinh x\]
for which \(y=1\) when \(x=\ln 3\).

Find the solution of the differential equation
\[x \frac{\mathrm{~d} y}{\mathrm{~d} x}+2 y=\mathrm{e}^{x}\]
for which \(y=3\) when \(x=1\). Give your answer in the form \(y=\mathrm{f}(x)\).

Find the solution of the differential equation
\[\frac{d y}{d x}+\frac{4 x^{3} y}{x^{4}+5}=6 x\]
for which \(y=1\) when \(x=1\). Give your answer in the form \(y=\mathrm{f}(x)\).

The linear transformation \(\mathrm{T}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) is represented by the matrix \(\mathbf{M}\), where
\(\mathbf{M}=\left(\begin{array}{rrrr} 3 & 2 & 0 & 1 \\ 6 & 5 & -1 & 3 \\ 9 & 8 & -2 & 5 \\ -3 & -2 & 0 & -1 \end{array}\right) .\)
(i) Find the rank of \(\mathbf{M}\).

Let \(K\) be the null space of T .
(ii) Find a basis for \(K\).

(iii) Find the general solution of
\(\mathbf{M} \mathbf{x}=\left(\begin{array}{r} 2 \\ 5 \\ 8 \\ -2 \end{array}\right) .\)

Find the value of \(a\) for which the system of equations
\(\begin{array}{l} x-y+2 z=4 \\ x+a y-3 z=b \\ x-y+7 z=13 \end{array}\)
where \(a\) and \(b\) are constants, has no unique solution.

Taking \(a\) as the value just found,
(i) find the general solution in the case \(b=-5\),

(ii) interpret the situation geometrically in the case \(b \neq-5\).

Find the value of \(a\) for which the system of equations
\(\begin{array}{l} x-y+2 z=4 \\ x+a y-3 z=b \\ x-y+7 z=13 \end{array}\)
where \(a\) and \(b\) are constants, has no unique solution.

Taking \(a\) as the value just found,
(i) find the general solution in the case \(b=-5\),

(ii) interpret the situation geometrically in the case \(b \neq-5\).