(a) Show that
\(\frac{\mathrm{d}}{\mathrm{~d} x}\left(\frac{x}{2} \sqrt{x^{2}-9}-\frac{9}{2} \cosh ^{-1} \frac{x}{3}\right)=\sqrt{x^{2}-9} .\)
(b) Find the solution of the differential equation
\(x \frac{\mathrm{~d} y}{\mathrm{~d} x}-y=x^{2} \sqrt{x^{2}-9}\)
given that \(y=1\) when \(x=3\). Give your answer in the form \(y=\mathrm{f}(x)\).

(a) Use the substitution \(u=1+x^{2}\) to find
\(\int \frac{x}{\sqrt{1+x^{2}}} \mathrm{~d} x\)
(b) Find the solution of the differential equation
\(x \frac{\mathrm{~d} y}{\mathrm{~d} x}-y=x^{2} \sinh ^{-1} x\)
given that \(y=1\) when \(x=1\). Give your answer in the form \(y=\mathrm{f}(x)\).

(a) Show that \(\frac{\mathrm{d}}{\mathrm{d} x}(\ln (\tanh x))=2 \operatorname{cosech} 2 x\).

(b) Find the solution of the differential equation
\(\sinh 2 x \frac{\mathrm{~d} y}{\mathrm{~d} x}+2 y=\sinh 2 x\)
for which \(y=5\) when \(x=\ln 2\). Give your answer in an exact form.