Question

Using properties of determinants, solve for x : $\begin{vmatrix}a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x\end{vmatrix}$ = 0

Solve the differential equation : $y+x{\frac{d y}{d x}}=x-y{\frac{d y}{d x}}.$

Find the particular solution of the differential equation : $y e^{y}\,d x=(y^{3}+2x e^{y})d y,y(0)=1$

Evaluate : $\large\textstyle\int{\frac{3x+1}{(x^{3}-x^{2}-x+1)}}d x\,.$

Using differentials, find the approximate value of $(3.968)^{1/2}$

If y(x) is a solution of the differential equation $\left(\frac{\dot{2}+\sin x}{1+y}\right)\frac{d y}{d x}\;=-\cos x\;\mathrm{and}\;y\;(0)=1,$ then find the value of $y\left({\frac{\pi}{2}}\right).$

Using properties of determinants, prove that following : $\begin{vmatrix}a+x & y & z \\ x & a+y & z \\ x & y & a+z\end{vmatrix}$ = $a^{2}(a+x+y+z).$