Question

Find the particular solution of the differential equation satisfying the given condition ${\frac{d y}{d x}}=y\tan x,$ given that y = 1, when x = 0.

If the function f : R → R be defined by f(x) = 2x – 3 and g : R → R by g(x) = x³ + 5, then find fog and show that fog is invertible. Also, find (fog)⁻¹, hence find (fog)⁻¹(9).

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.

Let $\stackrel{\rightarrow}{a}\$ = $\stackrel{\wedge}{i}+\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$, $\stackrel{\rightarrow}{b}\$ = $4\stackrel{\wedge}{i}-2\stackrel{\wedge}{j} +3\stackrel{\wedge}{k}$ , and $\stackrel{\rightarrow}{c}\$ = $\stackrel{\wedge}{i}-2\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$ Find a vector of magnitude 6 units,which is parallel to the vector $2\stackrel{\rightarrow}{a}-\stackrel{\rightarrow}{b}+3\stackrel{\rightarrow}{c}$.

Form the differential equation representing family of curves given by (x – a)² + 2y²= a², where a is an arbitrary constant.

Find the general solution of the following differential equation : $y-x{\frac{d y}{d x}}=a\left(y^{2}+{\frac{d y}{d x}}\right).$

For the curve y = 4x³ – 2x⁵, find all those points at which the tangent passes through the origin.