The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \sin \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right)\tan \left(\mathrm{x} + \mathrm{y}\right) - 1 \mathrm{is}$

• $\csc \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) + \tan \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) = \mathrm{x} + \mathrm{c}$

• $x +\hspace{0.17em}\csc \left(x + y\right) = c$

• $\mathrm{x} +\hspace{0.17em}\tan \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) = \mathrm{c}$

• $\mathrm{x} +\hspace{0.17em}\sec \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) = \mathrm{c}$

$$\begin{gathered} \mathrm{The} \mathrm{differential} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{family} \mathrm{y} = {\mathrm{ae}}^{\mathrm{x}} + {\mathrm{bxe}}^{\mathrm{x}} + {\mathrm{cx}}^2{\mathrm{e}}^{\mathrm{x}} \mathrm{of} \mathrm{curves}, \\ \mathrm{where} \mathrm{a}, \mathrm{b}, \mathrm{c} \mathrm{are} \mathrm{arbitrary} \mathrm{constants} , \mathrm{is} \end{gathered}$$

• $\mathrm{y}\text{'}\text{'}\text{'} + 3\mathrm{y}\text{'}\text{'} + 3\text{'} + \mathrm{y} = 0$

• $\mathrm{y}\text{'}\text{'}\text{'} + 3\mathrm{y}\text{'}\text{'} - 3\text{'} - \mathrm{y} = 0$

• $\mathrm{y}\text{'}\text{'}\text{'} - 3\mathrm{y}\text{'}\text{'} - 3\text{'} + \mathrm{y} = 0$

• $\mathrm{y}\text{'}\text{'}\text{'} - 3\mathrm{y}\text{'}\text{'} + 3\text{'} - \mathrm{y} = 0$

$\mathrm{The} \mathrm{solution} \mathrm{of} \tan \left(\mathrm{y}\right)\frac{\mathrm{dy}}{\mathrm{dx}} = \sin \left(\mathrm{x} + \mathrm{y}\right) + \sin \left(\mathrm{x} - \mathrm{y}\right) \mathrm{is}$

• sec(y) = 2cos(x) + c

• sec(y) = - 2cos(x) + c

• tan(y) =  - 2cos(x) + c

• sec²(y) = - 2cos(x) + c

A family of curves has the differential equation $\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 2{\mathrm{y}}^2 - {\mathrm{x}}^2$. Then, the family of curves is

• ${\mathrm{y}}^2 = {\mathrm{cx}}^2 + {\mathrm{x}}^3$

• ${\mathrm{y}}^2 = {\mathrm{cx}}^4 + {\mathrm{x}}^3$

• ${\mathrm{y}}^2 = \mathrm{x} + {\mathrm{cx}}^4$

• ${\mathrm{y}}^2 = {\mathrm{x}}^2 + {\mathrm{cx}}^4$

$$\begin{gathered} \mathrm{The} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \\ \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{y}}{\mathrm{x}} + \frac{\mathrm{\varphi }\left({\displaystyle \raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}\right)}{\mathrm{\varphi }\text{'}\left({\displaystyle \raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}\right)} \mathrm{is} \end{gathered}$$

• $\mathrm{x\varphi }\left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{k}$

• $\mathrm{\varphi }\left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{kx}$

• $\mathrm{y\varphi }\left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{k}$

• $\mathrm{\varphi }\left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{ky}$

If y = y(x) is the solution of the differential equation $\left(\frac{2 + \sin \left(\mathrm{x}\right)}{\mathrm{y} + 1}\right)\frac{\mathrm{dy}}{\mathrm{dx}} + \cos \left(\mathrm{x}\right) = 0$ then y($\frac{\mathrm{\pi }}{2}$)is equal to

• $\frac{1}{3}$

• $\frac{2}{3}$

• 1

• $\frac{4}{3}$

$$\begin{gathered} \mathrm{If} \mathrm{u} = \mathrm{f}\left(\mathrm{r}\right), \mathrm{where} {\mathrm{r}}^2 = {\mathrm{x}}^2 + {\mathrm{y}}^2, \mathrm{then} \\ \left(\frac{\partial \mathrm{u}}{\partial {\mathrm{x}}^2} + \frac{{\partial }^2\mathrm{u}}{\partial {\mathrm{y}}^2}\right) = ? \end{gathered}$$

• f''(r)

• f''(r) +f'(r)

• f''(r) + $\frac{1}{\mathrm{r}}$f'(r)

• f''(r) + rf'(r)

368.

$\mathrm{If} \frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{xtan}\left(\mathrm{x} - \mathrm{y}\right) = 1, \mathrm{then} \sin \left(\mathrm{x} - \mathrm{y}\right) = ?$

• ${\mathrm{Ae}}^{- {\mathrm{x}}^2}$

• Ae^2x

• ${\mathrm{Ae}}^{{\mathrm{x}}^2}$

• Ae ^{- 2x}

$$\begin{gathered} \mathrm{An} \mathrm{integrating} \mathrm{factor} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \\ \left(1 - {\mathrm{x}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{xy} = \frac{{\mathrm{x}}^4}{\left(1 + {\mathrm{x}}^5\right)}{\left(\sqrt{1 - {\mathrm{x}}^2}\right)}^3 \mathrm{is} \end{gathered}$$

• $\sqrt{1 - {\mathrm{x}}^2}$

• $\frac{\mathrm{x}}{\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{{\mathrm{x}}^2}{\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{1}{\sqrt{1 - {\mathrm{x}}^2}}$

$$\begin{gathered} \mathrm{If} {\cos }^{-1}\left(\frac{\mathrm{y}}{\mathrm{b}}\right) = 2\log \left(\frac{\mathrm{x}}{2}\right), \mathrm{where} \mathrm{x} > 0, \mathrm{then} \\ {\mathrm{x}}^2\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} = ? \end{gathered}$$

• 4y

• - 4y

• 0

• - 8y