For each of the following initial value problems verify that the accompanying functions is a solution. (i) `x(dy)/(dx)=1,\ y(1)=0 => y=logx` (ii) `(dy)/(dx)=y ,\ y(0)=1 => y=e^x` (iii) `(d^2y)/(dx^2)+y=0,\ y(0)=0,\ y^(prime)(0)=1 => y=sinx` (iv) `(d^2y)/(dx^2)-(dy)/(dx)=0,\ y(0)=2,\ y^(prime)(0)=1 => y=e^x+1` (v) `(dy)/(dx)+y=2,\ y(0)=3 => y=e^(-x)+2`

Explanation

(i)`y=log x `

`
x_{1}(f w cdot r cdot to x `

`
frac{d y}{d x}=frac{1}{x} `

`
x frac{d y}{d x}=1 `

`
y=log x `


Put ` x=1`

`
y(1)=log 1`

`
y(1)=0`

`
...