Mathematics : Differential Equations

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

If T = 2$\mathrm{\pi }$$\sqrt{\frac{1}{\mathrm{g}}}$, then relative errors in T and l are in the ratio

• 1/2

• 2

• 1/2$\mathrm{\pi }$

• None

The order of the differential equation whose general solution is given by

y = (C₁ + C₂)sin(x + C₃) - C₄e^{x + C₅}, is

• 2

• 3

• 4

• 5

The differential equation of the curve for which the initial ordinate of any tangent is equal to the corresponding subnormal

• is linear

• is homogeneous of second degree

• has separable variables

• is of second order

The solution of the equation $\frac{d\mathrm{y}}{d\mathrm{x}} = \cos \left(\mathrm{x} - \mathrm{y}\right)$ is

• $\mathrm{x} + \cot \left(\frac{\mathrm{x} - \mathrm{y}}{2}\right) = \mathrm{C}$

• $\mathrm{y} + \cot \left(\frac{\mathrm{x} - \mathrm{y}}{2}\right) = \mathrm{C}$

• $\mathrm{x} + \tan \left(\frac{\mathrm{x} - \mathrm{y}}{2}\right) = \mathrm{C}$

• None of these

The differential equation of all parabolas each of which has a latusrectum 4a and whose axes are parallel to the Y-axis is

• of order 1 and degree 2

• of order 2 and degree 3

• of order 2 and degree 1

• of order 2 and degree 2

The solution of $\frac{d^2\mathrm{x}}{d{\mathrm{y}}^2} - \mathrm{x} = \mathrm{k}$ where k is a non-zero constant, vanishes when y = 0 and tends of finite limit as y tends to infinity, is 

• x = k(1 + e^{- y})

• x = k(e^y + e^{- y} - 2)

• x = k(e^{- y} - 1)

• x = k(e^y - 1)

The differential equation (3x + 4y + 1)dx + (4x + 5y + 1)dy = 0 represents a family of

• circles

• parabolas

• ellipses

• hyperbolas

The solution of $\frac{d\mathrm{y}}{d\mathrm{x}} = \frac{{\mathrm{x}}^2 + {\mathrm{y}}^2 + 1}{2\mathrm{xy}}$, satisfying y(1) = 0 is given by

• hyperbola

• circle

• ellipse

• parabola

If $\mathrm{x} . \frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = \mathrm{x} . \frac{\mathrm{f}\left(\mathrm{xy}\right)}{\mathrm{f}\text{'}\left(\mathrm{xy}\right)}$, then f(xy) is equal to

• $\mathrm{k} . {\mathrm{e}}^{\frac{{\mathrm{x}}^2}{2}}$

• $\mathrm{k} . {\mathrm{e}}^{\raisebox{1ex}{${\mathrm{y}}^2$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

• $\mathrm{k} . {\mathrm{e}}^{x^2}$

• $\mathrm{k} . {\mathrm{e}}^{\frac{\mathrm{xy}}{2}}$