Question

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.

Find the particular solution of the differential equation : ${\frac{d y}{d x}}\ +y\;{\cot{x}}=$ 4x cosec x, (x ≠ 0), given, that y = 0,when $x={\frac{\pi}{2}}$

If A = $\left(\begin{array}{l l l }2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0\end{array}\right)$ , find A² – 5A + 4I and hence find a matrix X such that A² – 5A + 4I + X = 0.

Solve the following differential equation. $(x^{3}+x^{2}+x+1){\frac{d y}{d x}}=2x^{2}+x.$

Find the equation of tangent to the curve x² + 3y = 3, which is parallel to line y – 4x + 5 = 0.

Find the equations of the normal to the curve y = 4x³ – 3x + 5 which are perpendicular to the line 9x – y + 5 = 0..

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