Differential Equations

The order and degree of the differential equation $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} = \sqrt[3]{1 - {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^4}$ are respectively

• 2, 3

• 3, 2

• 2, 4

• 2, 2

Form the differential equation of all family of lines y = mx ± $\frac{4}{\mathrm{m}}$ eliminating the arbitrary m constant 'm' is

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

• $\mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) - \mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}} +\hspace{0.17em}4 = 0$

• $\mathrm{x}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + \mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}} +\hspace{0.17em}4 = 0$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

The differential equation of family of circles whose centre lies on x-axis, is

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + 1 = 0$

• $\mathrm{y}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 - 1 = 0$

• $\mathrm{y}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 - 1 = 0$

• $\mathrm{y}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + 1 = 0$

The solution of the differential equation $\mathrm{y}\left(1 + \log \left(\mathrm{x}\right)\right)\frac{d\mathrm{y}}{d\mathrm{x}} - \mathrm{xlog}\left(\mathrm{x}\right) = 0$ is

• x log(x) = y + c

• x log(x) = yc

• y(1 + log(x) = c

• log(x) - y = c

The order of the differential equation whose solution is ae^x + be^2x + ce^3x + d = 0, is

• 1

• 2

• 3

• 4

The differential equation of all circles which pass through the origin and whose centres lie on y-axis is

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

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

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

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

If m and n are order and degree of the equation

${\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^5 + 4 . \frac{{\left({\displaystyle \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}}\right)}^3}{\left({\displaystyle \frac{{\mathrm{d}}^3\mathrm{y}}{{\mathrm{dx}}^3}}\right)} + \left(\frac{{\mathrm{d}}^3\mathrm{y}}{{\mathrm{dx}}^3}\right) = {\mathrm{x}}^2 - 1$, then

• m = 3, n = 3

• m = 3, n = 2

• m = 3, n = 5

• m = 3, n = 1

The integrating factor of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}\left(\mathrm{xlog}\left(\mathrm{x}\right)\right) + \mathrm{y} = 2\log \left(\mathrm{x}\right)$ is iven by

• e^x

• log(x)

• log(log(x))

• x

The differential equation whose solution is (x - h)² + (y - k)² = a² (a is a constant), is

• ${\left[1+{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]}^3 = {\mathrm{a}}^2\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$

• ${\left[1+{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]}^3 = {\mathrm{a}}^2{\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^2$

• ${\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)\right]}^3 = {\mathrm{a}}^2{\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^2$

• None of these

The differential equation of the family of circles touching Y-axis at the origin is

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

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

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

• $\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{xy} = 0$