Question

Solve the differential equation: $x\,{\frac{d y}{d x}}\ +y=x\cos x+\sin x,$ given $y\left({\frac{\pi}{2}}\right)=1.$

Find : $\large\textstyle\int$$\large\frac{x^{2}\,d x}{x^{4}+x^{2}-12}\;.$

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls ? Given that : (i) the youngest is a girl. (ii) atleast one is a girl.

Using properties of determinants, prove that : $\begin{vmatrix}b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y\end{vmatrix}$ = 2 $\begin{vmatrix}a & b & c \\ p & q & r \\ x & y & z\end{vmatrix}$ .

Solve the differential equation $(x^{2}\ -\ y x^{2})d y$ + $(y^{2}+x^{2}y^{2})d x=0,$ given that y = 1 when x = 1.

Using properties of determinants, prove that : $\begin{vmatrix}x²+1 & xy & xz \\ xy & y²+1 & yz \\ xz & yz & z²+1\end{vmatrix}$

Solve the differential equation $(\tan^{-1}x-y)d x=(1+x^{2})\,d y$