Find the second solution of the differential equation \( t^{2} y^{\prime \prime}-2 t y^{\prime}+2 y=0 \) if \( y_{1}=t^{2}, t>0 \) Hence find the general solution of the differential equation.

Find the particular solution for the differential equation \( \frac{dy}{dx} = y \sin(x) \) given that \( y(0) = 1 \).

\( ( x ^ { p } ) ^ { \prime } = \)

Find the integration distance \( \langle f, g\rangle=\int_{0}^{1} f(x) g(x) d x \) between the given polynomials. \( f(x)=8 x-3, g(x)=2 x-7 \)

2 Sketch the graphs of \( y=\ln (x+1) \) and \( y=-2 x-2 \) on the same axes. Hence, show that the equation \( \ln (x+1)+2 x+2=0 \) has only one real root. By using Newton-Raphson formulae, deduce the iterative equation of the root is \( x_{n+1}=x_{n}-\frac{\left(x_{n}+1\right)\left[\ln \left(x_{n}+1\right)+2 x_{n}+2\right]}{3+2 x_{n}} \). Hence, find the real root correct to four decimal places with initial value \( x_{0}=-0.1 \).

6.10 Calcolare la somma delle seguenti serie geometriche \( \begin{array}{ll}\text { (a) } \sum_{n=0}^{\infty} \frac{1}{2^{n}} & \text { (b) } \sum_{n=1}^{\infty} \frac{1}{2^{n}} \\ \text { (c) } \sum_{n=0}^{\infty} 3^{n} & \text { (d) } \sum_{n=0}^{\infty}\left(\frac{4}{5}\right)^{n}\end{array} \)