What is the general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{\mathrm{x}}{\mathrm{y}} = 0 ?$

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

• ${\mathrm{x}}^2 - {\mathrm{y}}^2 = \mathrm{c}$

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

• x + y = c

$\mathrm{The} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \frac{\mathrm{dy}}{\mathrm{dx}} = \cos \left(\mathrm{y} - \mathrm{x}\right) + 1 \mathrm{is} :$

• ${\mathrm{e}}^{\mathrm{x}}\left[\sec \left(\mathrm{y} - \mathrm{x}\right) - \tan \left(\mathrm{y} - \mathrm{x}\right)\right] = \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\left[\sec \left(\mathrm{y} - \mathrm{x}\right) + \tan \left(\mathrm{y} - \mathrm{x}\right)\right] = \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\sec \left(\mathrm{y} - \mathrm{x}\right)\tan \left(\mathrm{y} - \mathrm{x}\right) = \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}} = \mathrm{c}\left(\sec \left(\mathrm{y} - \mathrm{x}\right) + \tan \left(\mathrm{y} - \mathrm{x}\right)\right)$

$\mathrm{If} \mathrm{y} = \mathrm{acos}\left(2\mathrm{x}\right) + \mathrm{bsin}\left(2\mathrm{x}\right), \mathrm{then} :$

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

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

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} -4\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 4\mathrm{y} = 0$

The differential equation of the system of circles touching the y-axis at the origin is : 

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 +\hspace{0.17em}2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• ${\mathrm{x}}^2 - {\mathrm{y}}^2 + 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• ${\mathrm{x}}^2 - {\mathrm{y}}^2 - 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

5.

Consider the following in respect of the differential equation : 

$\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 2{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + 9\mathrm{y} = \mathrm{x}$

1. The degree of the differential equation is 1.
2. The order of the differential equation is 2.

Which of the above statements is/are correct?

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

What is the solution xdy - ydx = 0 ?

• xy = c

• y = cx

• x + y = c

• x - y = c

The order and degree of the differential equation y²  = 4a(x - a), where 'a'· is an arbitrary constant, are respectively : 

• 1, 2

• 2, 1

• 2, 2

• 1, 1

Which one of the following differential equations has a periodic solution ?

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

• $\frac{{\mathrm{d}}^2\mathrm{x}}{{\mathrm{dt}}^2} - \mathrm{\mu x} = 0$

• $\mathrm{x}\frac{\mathrm{dx}}{\mathrm{dt}} + \mathrm{\mu t} = 0$

• $\frac{\mathrm{dx}}{\mathrm{dt}} + \mathrm{\mu xt} = 0$

$\mathrm{If} \mathrm{y} = {\mathrm{e}}^{{\mathrm{x}}^2}\sin \left(2\mathrm{x}\right), \mathrm{then} \mathrm{what} \mathrm{is} \frac{\mathrm{dy}}{\mathrm{dx}} \mathrm{at} \mathrm{x} = \mathrm{\pi } \mathrm{equal} \mathrm{to} ?$

• $\left(1 + \mathrm{\pi }\right){\mathrm{e}}^{{\mathrm{\pi }}^2}$

• $2\mathrm{\pi }{\mathrm{e}}^{{\mathrm{\pi }}^2}$

• $2{\mathrm{e}}^{{\mathrm{\pi }}^2}$

• ${\mathrm{e}}^{{\mathrm{\pi }}^2}$

What is the solution of :

(1 + 2x)dy - (1 - 2y)dx = 0 ?

• x - y - 2xy = c

• y - x - 2xy = c

• y + x - 2xy = c

• x + y + 2xy = c