Question

Find the solution of the differential equation ${\frac{d y}{d x}}=x^{3}e^{-2y}$.

Find the sum of the vectors : $\stackrel{\rightarrow}{a}=\stackrel{\wedge}{i}-2\stackrel{\wedge}{j},$ $\vec{b}=2\hat{i}-3\hat{j},$ $\vec{c}=2\hat{i}+3\hat{k},$

Write the value of x + y + z if $\left(\begin{array}{l l l }1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$ $\left(\begin{array}{l }x \\ y \\ z\end{array}\right)$ = $\left(\begin{array}{l }1 \\ -1 \\ 0\end{array}\right)$.

Evaluate $\textstyle\int\!{\frac{x^{3}-x^{2}+x-1}{x-1}}\,d x\,.$

If $\left(\begin{array}{l l l }1 & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & 1\end{array}\right)$ $\left(\begin{array}{l }x \\ -1 \\ z\end{array}\right)$ = $\left(\begin{array}{l }1 \\ 2 \\ 1\end{array}\right)$ , find x + y + z.

The two vectors $:{\hat{j}}+{\hat{k}}$ and $3\stackrel{\wedge}{i}-\stackrel{\wedge}{j} +4\stackrel{\wedge}{k}$ represent the two sides AB and AC, respectively of $\Delta A B C.$ Find the length of the median through A.

Write the value of $\textstyle\int\!{\frac{\ d x}{x^{2}+16}}$