Question

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

If A = $\left(\begin{array}{l l }1 & -1 \\ 2 & -1\end{array}\right)$ , B = $\left(\begin{array}{l l }a & 1 \\ b & -1\end{array}\right)$ and (A + B)² = A²+ B², then find the values of a and b.

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 $2\stackrel{\rightarrow}{a}-\stackrel{\rightarrow}{b}+3\stackrel{\rightarrow}{c}$.

Solve $\left(1+x^{2}\right){\frac{d y}{d x}}+2x y-4x^{2}=0$ subject to the initial condition y(0) = 0.

Find the particular solution of the following differential equation. $\cos y\,d x+(1+2e^{-x})\sin y\,d y=0;y(0)={\frac{\pi}{4}}$

Solve the following differential equation : $\left(1+e^{\frac{x}{y}}\right)d x+e^{\frac{x}{y}}\left(1-{\frac{x}{y}}\right)d y=0.$

Find : $\large\textstyle\int$$\frac{x^{4}+1}{x\left(x^{2}+1\right)^{2}}d x$.