(a) [6 marks] Find the general solution of the differential equations (i) \( x \cos ^{4} y d x+\tan y d y=0 \) (ii) \( (\cos x \cos y-\cot x) d x-\sin x \sin y d y=0 \) (iii) \( y^{\prime \prime}-6 y^{\prime}+9 y=8 e^{5 x}+100 \sin x \) (b) \( [6 \) marks \( ] \) If \( F(s)=\mathcal{L}(f(t)) \), prove that \[ \mathcal{L}(t f(t))=-\frac{d F}{d s} \]

Let \( y-x e^{y}=1 \), then \( d y(0)= \)

\( \left. \begin{array} { l }{ \frac { d } { d x } ( x ^ { 2 } \sin ^ { 2 } x ^ { 2 } ) } \\ { = x ^ { 2 } \frac { d } { d x } ( \sin ^ { 2 } x ^ { 2 } ) + \sin ^ { 2 } x ^ { 2 } \frac { d } { d x } ( x ^ { 2 } ) } \\ { = x ^ { 2 } \frac { d } { d x } ( \sin x ) ^ { 2 } + \sin ^ { 2 } x ^ { 2 } \frac { d } { d x } ( x ^ { 2 } ) } \\ { = x ^ { 2 } ( 2 \cos x ) + \sin ^ { 2 } x ^ { 2 } 2 x } \\ { = 2 x ^ { 2 } \cos x + 2 x \sin ^ { 2 } x ^ { 2 } } \end{array} \right. \)

\( \left. \begin{array} { l } { \frac { d } { d x } ( \sin ^ { 2 } x ^ { 2 } ) } \\ { \frac { d } { d x } ( \sin x ) ^ { 2 } } \\ { = 2 \cos x } \end{array} \right. \)

22. a) If \( f(x)=\frac{|6-x|}{x-6} \), find \( \lim _{x \rightarrow 6+} f(x) \) (if exist.) b) Evaluate: \( \lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x^{3}} \) c) Evaluate: \( \lim _{x \rightarrow 0} \frac{x\left(2^{2}-1\right)}{1-\cos x} \) KMC Note d) Find \( \frac{d x}{d y} \) if \( x^{3}+y^{3}=3 a x^{2} y \) e) What is the derivative of \( \sec ^{-1} x \) with respect to ' \( x \) '.

4. (15б) Выясните, является ли функция \( z=\ln \left(x^{2}+y^{2}+2 y+1\right) \) решением уравнения \( \frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0 \)