The differential equation of the family of curves y = e^2x(acos(x) + bsin(x)), where a and b are arbitrary constants, is given by

• y₂ - 4y₁ + 5y = 0

• 2y₂ - y₁ + 5y = 0

• y₂ + 4y₁ - 5y = 0

• y₂ - 2y₁ + 5y = 0

The solution of the differential equation $\mathrm{xdy} - \mathrm{ydx} = \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2}\mathrm{dx}$ is

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

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

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

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

The differential equation of all non-vertical lines in a plane is

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} = 0$

• $\frac{d^2\mathrm{x}}{d{\mathrm{y}}^2} = 0$

• $\frac{d\mathrm{y}}{d\mathrm{x}} = 0$

• $\frac{d\mathrm{x}}{d\mathrm{y}} = 0$

The differential equation of all parabolas whose axes are parallel to y-axis is

• $\frac{d^3\mathrm{y}}{d{\mathrm{x}}^3} = 0$

• $\frac{d^2\mathrm{x}}{d{\mathrm{y}}^2} = 0$

• $\frac{d^3\mathrm{y}}{d{\mathrm{x}}^3}+ \frac{d^2\mathrm{x}}{d{\mathrm{y}}^2} = 0$

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + 2\frac{d\mathrm{y}}{d\mathrm{x}} = 0$

Solution of $$\begin{gathered} \mathrm{Given} \mathrm{equation} \mathrm{is} \frac{d\mathrm{y}}{d\mathrm{x}} = \frac{\mathrm{xlog}\left({\mathrm{x}}^2\right) + \mathrm{x}}{\sin \left(\mathrm{y}\right){\displaystyle }{\displaystyle +}{\displaystyle }{\displaystyle \mathrm{ycos}}{\displaystyle \left(\mathrm{y}\right)}} \\ \Rightarrow \left(\sin \left(\mathrm{y}\right) + \mathrm{ycos}\left(\mathrm{y}\right)\right)\mathrm{dy} = \left(\mathrm{xlog}\left({\mathrm{x}}^2\right) + \mathrm{x}\right)\mathrm{dx} \\ \mathrm{On} \mathrm{integrating} \mathrm{both} \mathrm{sides}, \mathrm{we} \mathrm{get} \\ \int \left(\sin \left(\mathrm{y}\right) + \mathrm{ycos}\left(\mathrm{y}\right)\right)\mathrm{dy} = \int \left(\mathrm{xlog}\left({\mathrm{x}}^2\right) + \mathrm{x}\right)\mathrm{dx} \\ \Rightarrow - \cos \left(\mathrm{y}\right) + \mathrm{ysin}\left(\mathrm{y}\right) + \cos \left(\mathrm{y}\right) = \frac{{\mathrm{x}}^2}{2}\log \left({\mathrm{x}}^2\right) \end{gathered}$$ is

• $\mathrm{ysin}\left(\mathrm{y}\right) = {\mathrm{x}}^2\log \left(\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{ysin}\left(\mathrm{y}\right) = {\mathrm{x}}^2 + \mathrm{C}$

• $\mathrm{ysin}\left(\mathrm{y}\right) = {\mathrm{x}}^2 + \log \left(\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{ysin}\left(\mathrm{y}\right) = \mathrm{xog}\left(\mathrm{x}\right) + \mathrm{C}$

66.

If x^py^q = (x + y)^{p + q}, then $\frac{d\mathrm{y}}{d\mathrm{x}}$ is equal to

• $\frac{\mathrm{y}}{\mathrm{x}}$

• $\frac{\mathrm{py}}{\mathrm{qx}}$

• $\frac{\mathrm{x}}{\mathrm{y}}$

• $\frac{\mathrm{qy}}{\mathrm{px}}$

The solution of $\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{ytan}\left(\mathrm{x}\right) = \sec \left(\mathrm{x}\right)$ is

• $\mathrm{ysec}\left(\mathrm{x}\right) = \tan \left(\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{ytan}\left(\mathrm{x}\right) = \sec \left(\mathrm{x}\right) + \mathrm{C}$

• $\tan \left(\mathrm{x}\right) = \mathrm{ytan}\left(\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{xsec}\left(\mathrm{x}\right) = \mathrm{ytan}\left(\mathrm{y}\right) + \mathrm{C}$

If y = ${\tan }^{-1}\left[\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)}\right]$, then $\frac{d\mathrm{y}}{d\mathrm{x}}$ is

• $\frac{1}{2}$

• $\frac{\mathrm{\pi }}{4}$

• 0

• 1

The differential equation $\frac{d\mathrm{y}}{d\mathrm{x}} + \frac{{\mathrm{y}}^2}{{\mathrm{x}}^2} = \mathrm{y}/\mathrm{x}$ has the solution

• x = y(log(x) + C)

• y = x(log(y) + C)

• x = (y + C)log(x)

• y = (x + C)log(y)

The degree of the differential equation satisfying $\sqrt{1 - {\mathrm{x}}^2} + \sqrt{1 - {\mathrm{y}}^2} =\mathrm{a}\left(\mathrm{x} - \mathrm{y}\right)$

• 1

• 2

• 3

• None