Question

Solve the following differential equation : ${\frac{d y}{d x}}={\frac{y}{x}}+{\frac{\sqrt{x^{2}+y^{2}}}{x}},\,x>0\,.$

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

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

Evaluate : $\large\textstyle\int\!{\frac{\ d x}{x(x^{5}+3)}}\,.$

Solve $\left(1+x^{2}\right){\frac{d y}{d x}}+2x y-4x^{2}=0$ subject to the initial condition y(0) = 0.

Find the particular solution of the differential equation $e^{x}{\sqrt{1-y^{2}}}d x+\left({\frac{y}{x}}\right)d x=0$ given that y = 1 when x = 0.

If A = $\left(\begin{array}{l l }2 & 3 \\ 1 & -4\end{array}\right)$ , B = $\left(\begin{array}{l l }1 & -2 \\ -1 & 3\end{array}\right)$ , verify that $(A B)^{-1}=B^{-1}A^{-1}$