Question

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

Solve the following differential equation. (1 + y²) (1 + log |x|) dx + x dy = 0

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing, when the foot of the ladder is 4 m away from the wall ?

Can $y=a x+{\frac{b}{a}}$ be a solution of the following differential equation ? $y=x{\frac{d y}{d x}}+{\frac{b}{\frac{d y}{d x}}}$ If no, find the solution of the D.E.

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

Show that the relation R on the set Z of all integers defined by (x, y) ∈ $R\Leftrightarrow(x-y)$ is divisible by 3 is an equivalence relation.

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