Question

Find the particular solution of the following differential equation given that at x = 2, y = 1 $x\,{\frac{d y}{d x}}\,+\,2y=x^{2},\,(x\,\neq\,0).$

Let f, g : R → R be two functions defined as f (x) = |x| + x and g (x) = |x| – x for all x ∈ R.Then find fog and gof.

Find the position vector of a point R, which divides the line joining two points P and Q whose position vectors are $2\stackrel{\rightarrow}{a}+\stackrel{\rightarrow}{b}$ and $\stackrel{\rightarrow}{a}-3\stackrel{\rightarrow}{b}$ respectively, externally in the ratio 1 : 2 Also, show that P is the mid-point, of line segment RQ.

Find the equations of tangents to the curve y = (x²– 1) (x – 2) at the points, where the curve cuts the X-axis.

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.

Find the points on the curve x² + y² – 2x – 3 = 0 at which tangent is parallel to x-axis.

Find the equations of the normal to the curve y = x³+ 2x + 6, which are parallel to line x + 14y + 4 = 0.