Question $1(10$ marks$)$

i$.$ Prove that $cosh(nx)sinh(nx)=[cosh(x)sinh(x){]}^n,$ for any positive integral value $n.$

ii$.$ Evaluate

${\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}{x}^{3}cos(x)dx$

iii. By making the substitution $x=sinh(),$ evaluate:

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^{2}1}{x-\sqrt[\phantom{2}]{x^{2}1}}dx$

leaving your answer in terms of the substituted variable.

$[5]$

Question $2(20$ marks$)$

i$.$ By making the substitution $y=vx$ and $y^{\text{'}}=vx{v}^{\text{'}},$ prove

that $y=x\sqrt[\phantom{2}]{(8x1)}$ solves $2xy{y}^{\text{'}}{x}^2-3y^2=0,$ given $y(1)=3.$

ii$.$ Solve $y^{\text{'}\text{'}}8y^{\text{'}}41y=0,y(0)=0,y^{\text{'}}(0)=15.$

iii. Solve the differential equation $y^{\text{'}\text{'}}-5y^{\text{'}}6y=x^2.$