Differential Equations

Degree and order of the differential equation $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} = {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2$ are respectively

• 1, 2

• 2, 1

• 2, 2

• 1, 1

The solution of the differential equation (1 + y) tan^-1(x)dx + y(1 + x²)dy = 0 is

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

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

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

• $\left({\tan }^{-1}\left(\mathrm{x}\right)\right)\left(1 + {\mathrm{y}}^2\right) + \mathrm{c} = 0$

An integrating factor of the differential equation

$\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} +\hspace{0.17em}\mathrm{ylog}\left(\mathrm{x}\right) = {\mathrm{xe}}^{\mathrm{x}}{\mathrm{x}}^{\frac{1}{2}\log \left(\mathrm{x}\right)}$, (x > 0), is

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

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

• ${\left(\sqrt{\mathrm{e}}\right)}^{{\left(\log \left(\mathrm{x}\right)\right)}^2}$

• ${\mathrm{e}}^{{\mathrm{x}}^2}$

Solution of the differential equation

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

• $\sec \left(\mathrm{y}\right) + 2\cos \left(\mathrm{x}\right) = \mathrm{c}$

• $\sec \left(\mathrm{y}\right) - 2\cos \left(\mathrm{x}\right) = \mathrm{c}$

• $\cos \left(\mathrm{y}\right) - 2\sin \left(\mathrm{x}\right) = \mathrm{c}$

• $\tan \left(\mathrm{y}\right) - 2\sec \left(\mathrm{y}\right) = \mathrm{c}$

The order of the differential equation whose solution is y = a cos(x) + b sin(x) + ce^-x , is

• 3

• 1

• 2

• 4

The differential equation of all parabolas with axis parallel to the axis of y is

• y₂ = 2y₁ + x

• y₃ = 2y₁

• ${\mathrm{y}}_2^3$ = y₁

• None of these

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{x} - \mathrm{y} + 3}{2\left(\mathrm{x} - \mathrm{y}\right) + 5}$ is

• $2\left(\mathrm{x} - \mathrm{y}\right) + \log \left(\mathrm{x} - \mathrm{y}\right) = \mathrm{x} \hspace{0.17em}+ \mathrm{c}$

• $2\left(\mathrm{x} - \mathrm{y}\right) - \log \left(\mathrm{x} - \mathrm{y} + 2\right) = \mathrm{x} \hspace{0.17em}+ \mathrm{c}$

   

• $2\left(\mathrm{x} - \mathrm{y}\right) + \log \left(\mathrm{x} - \mathrm{y} + 2\right) = \mathrm{x} \hspace{0.17em}+ \mathrm{c}$

• None of the above

The solution of the differential equation (3.xy + y²)dx + (x² + xy)dy = 0 is

• x²(2xy + y²) = c²

• x²(2xy - y²) = c²

• x²(y² - 2xy) = c²

• None of these

The order and degree of the differential equation $\sqrt{\frac{\mathrm{dy}}{\mathrm{dx}}} - 4\frac{\mathrm{dy}}{\mathrm{dx}}$ - 7x = 0 are

• 1 and 1/2

• 2 ana 1

• 1 and 1

• 1 and 2

General solution of the differential equation 

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{x} + \mathrm{y} + 1}{\mathrm{x} + \mathrm{y} - 1}$ is given by

• $\mathrm{x} + \mathrm{y} = \log \left|\mathrm{x} + \mathrm{y}\right| + \mathrm{c}$

• $\mathrm{x} - \mathrm{y} = \log \left|\mathrm{x} + \mathrm{y}\right| + \mathrm{c}$

• $\mathrm{y} = x + \log \left|\mathrm{x} + \mathrm{y}\right| + \mathrm{c}$

• $\mathrm{y} = \mathrm{xlog}\left|\mathrm{x} + \mathrm{y}\right| + \mathrm{c}$