Question

Solve the following differential equation. (1 + y²) (1 + log |x|) dx + x dy = 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 $2\stackrel{\rightarrow}{a}-\stackrel{\rightarrow}{b}+3\stackrel{\rightarrow}{c}$.

Find : $\textstyle\int$${\frac{x^{2}}{x^{4}+x^{2}-2}}d x$

Solve the differential equation $(x^{2}\ -\ y x^{2})d y$ + $(y^{2}+x^{2}y^{2})d x=0,$ given that y = 1 when x = 1.

Consider f :$R_{+}\to[4,\infty)$ given by f(x) = x² + 4 Show that f is invertible with the inverse $f^{-1}$ of f given by $f^{-1}(y)$ = $\sqrt{y-4}$ , where $\textstyle R_{+}$ is the set of all nonnegative real numbers.

Find the points on the curve y = x³ at which the slope of the tangent is equal to y-coordinate of the point.

Find the equations of the tangent and the normal, to the curve 16x² + 9y² = 145 at the point $(x_{1},\;y_{1}),$ where x₁ = 2 and y₁ > 0.