Question

Find the particular solution of the following differential equation : $(x\ +1){\frac{d y}{d x}}=2e^{-y}-1;$ y = 0 when x = 0.

Find : $\large\textstyle\int\!{\large\frac{x^{2}+x+1}{(x^{2}+1)\left(x+2\right)}}\,d x.$

Solve the following differential equation : $x{\frac{d y}{d x}}\ +y-x+x y\cot x=0,x\neq0$

Find the particular solution of the following differential equation : $x\left(x^{2}-1\right){\frac{d y}{d x}}\ =1;$ y = 0, when x = 2.

A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game, then find the probability that B wins.

Can $y=a x+{\frac{b}{a}}$ be a solution of the following differential equation ? $y=x{\frac{d y}{d x}}+{\frac{b}{\frac{d y}{d x}}}$ If no, find the solution of the D.E.

Find : $\large\textstyle\int$$\large{\frac{x}{(x^{2}+1)(x-1)}}\,d x.$