Question

Write the order of the differential equation: $\log\,\left(\frac{d^{2}y}{d x^{2}}\right)=\left(\frac{d y}{d x}\right)^{3}+x$

Find the sum of the vectors $\stackrel{\rightarrow}{a}=\hat{i}-2\hat{j}+\hat{k}\;,$ $\vec{b}=-2\hat{i}+4\hat{j}+5\hat{k}$ and $\stackrel{\rightarrow}{c}=\hat{i}-6\hat{j}-7\hat{k}.$

If x $\left(\begin{array}{l }2 \\ 3\end{array}\right)$ + y $\left(\begin{array}{l }-y \\ y\end{array}\right)$ = $\left(\begin{array}{l }10 \\ 5\end{array}\right)$ , write the value of x.

Write the differential equation obtained by eliminating the arbitrary constant C in the equation representing the family of curves xy = C cos x.

L and M are two points with position vectors $2\stackrel{\rightarrow}{a}-\stackrel{\rightarrow}{b}$ and $\stackrel{\rightarrow}{a}+2\stackrel{\rightarrow}{b}$ respectively. Write the position vectors of a point N which divides the line segment LM in the ratio 2:1 externally.

Write a unit vector in the direction of the sum of vectors : $\stackrel{\rightarrow}{a}\ = 2\stackrel{\wedge}{i}-\stackrel{\wedge}{j} +2\stackrel{\wedge}{k} and \stackrel{\rightarrow}{b}\ = \stackrel{\wedge}{i}+\stackrel{\wedge}{j} +3\stackrel{\wedge}{k}$

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