Question

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 following differential equation. $\cos y\,d x+(1+2e^{-x})\sin y\,d y=0;y(0)={\frac{\pi}{4}}$

Using differentials, find the approximate value of ${\sqrt{49\cdot5}}\ .$

Two farmers X and Y cultivate only three varieties of rice namely Basmati, Permal and Naura. The sale in rupees of these varieties of rice by both the farmers in the months of September and October are given by the following matrices A and B.

Using properties of determinants, prove that : $\begin{vmatrix}a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b\end{vmatrix}$ = $2(a+b+c)^{3}.$

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

Find the particular solution of the differential equation x (1 + y²) dx – y (1 + x²) dy = 0, given that y = 1, when x =0.