Mathematics : Differential Equations

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

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

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

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

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

The degree and order of the differential equation ${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^3\right]}^{\frac{7}{3}} = 7\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)$ respectively are

• 3 and 7

• 3 and 2

• 7 and 3

• 2 and 3

The particular solution of the differential equation

$\mathrm{y}\left(1 + \log \left(\mathrm{x}\right)\right)\frac{d\mathrm{x}}{d\mathrm{y}} - \mathrm{xlog}\left(\mathrm{x}\right) = 0$, when, x = e, y = e² is

• y = exlog(x)

• ey = xlog(x)

• xy = elog(x)

• ylog(x) = ex

If $\sin \left(\mathrm{x}\right)$  is the integrating factor (IF) of the linear differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{Py} = \mathrm{Q}$, then P is

• $\log \left(\sin \left(\mathrm{x}\right)\right)$

• $\cos \left(\mathrm{x}\right)$

• $\tan \left(\mathrm{x}\right)$

• $\cot \left(\mathrm{x}\right)$

The solution of the differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}} = \tan \left(\frac{\mathrm{y}}{\mathrm{x}}\right) + \frac{{\displaystyle \mathrm{y}}}{{\displaystyle \mathrm{x}}}$ is

• $\cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{cx}$

• $\sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{cx}$

• $\cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{cy}$

• $\sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{cy}$

The differential equation of all parabolas whose axis is Y-axis, is

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

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

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

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

The particular solution of the differential equation xdy + 2ydx = 0, when x = 2, y = 1 is

• xy = 4

• x²y = 4

• xy² = 4

• x²y² = 4

The general solution of the equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{\mathrm{y}}^2 - \mathrm{x}}{2\mathrm{y}\left(\mathrm{x} + 1\right)}$ is

• y² = (1 + x)log(1 + x) - c

• ${\mathrm{y}}^2 = \left(1 + \mathrm{x}\right)\log \left(\frac{\mathrm{c}}{\left(1 - \mathrm{x}\right)}\right) - 1$

• ${\mathrm{y}}^2 = \left(1 - \mathrm{x}\right)\log \left(\frac{\mathrm{c}}{\left(1 - \mathrm{x}\right)}\right) - 1$

• ${\mathrm{y}}^2 = \left(1 + \mathrm{x}\right)\log \left(\frac{\mathrm{c}}{\left(1 + \mathrm{x}\right)}\right) - 1$

The general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \sin \left(\frac{\mathrm{x} + \mathrm{y}}{2}\right) = \sin \left(\frac{{\displaystyle \mathrm{x} - \mathrm{y}}}{{\displaystyle 2}}\right)$ is

• ${\log }_{\mathrm{e}}\left|\tan \left(\frac{\mathrm{y}}{2}\right)\right| = - 2\sin \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{C}$

• ${\log }_{\mathrm{e}}\left|\tan \left(\frac{\mathrm{y}}{4}\right)\right| = 2\sin \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{C}$

• ${\log }_{\mathrm{e}}\left(\tan \left(\frac{\mathrm{y}}{2}\right)\right) = - 2\sin \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{C}$

• ${\log }_{\mathrm{e}}\left(\tan \left(\frac{\mathrm{y}}{4}\right)\right) = - 2\sin \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{C}$

The function y specified implicitly by the relation ${\int }_0^{\mathrm{y}}{\mathrm{e}}^{\mathrm{t}}\mathrm{dt} + {\int }_0^{\mathrm{x}}\cos \left(\mathrm{t}\right)\mathrm{dt}$ = 0 satisfies the differential equation

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

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

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

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