21.

If $\sqrt{\mathrm{y}} = {\cos }^{-1}\left(\mathrm{x}\right)$, then it satisfies the differential equation

$\left(1 - {\mathrm{x}}^2\right)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} - \mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}} = \mathrm{c}$, where c equal to 

• 0

• 3

• 1

• 2

The solution of the differential equation $\mathrm{y}\frac{d\mathrm{y}}{d\mathrm{x}} = \mathrm{x}\left[\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2} + \frac{\mathrm{\varphi }\left(\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}\right)}{\mathrm{\varphi }\text{'}\left(\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}\right)}\right]$ is (where, c is a constant)

• $\mathrm{\varphi }\left(\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}\right) = \mathrm{cx}$

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

• $\mathrm{\varphi }\left(\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}\right) = {\mathrm{cx}}^2$

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

The curve $\mathrm{y} = {\left(\cos \left(\mathrm{x}\right) \hspace{0.17em}+ \mathrm{y}\right)}^{1/2}$ satisfies the differential equation

• $\left(2\mathrm{y} - 1\right)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + 2{\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^2 +\hspace{0.17em} \cos \left(\mathrm{x}\right) = 0$

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

• $\left(2\mathrm{y} - 1\right)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} - 2{\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^2 +\hspace{0.17em} \cos \left(\mathrm{x}\right) = 0$

• $\left(2\mathrm{y} - 1\right)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} - {\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^2 + \hspace{0.17em}\cos \left(\mathrm{x}\right) = 0$

The solution of the differential equation

$\frac{d\mathrm{y}}{d\mathrm{x}} + \frac{\mathrm{y}}{{\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{x}\right)} = \frac{1}{\mathrm{x}}$

under the condition y = 1 when x = e is

• $2\mathrm{y} = {\log }_{\mathrm{e}}\left(\mathrm{x}\right) +\hspace{0.17em}\frac{1}{{\log }_{\mathrm{e}}\left(\mathrm{x}\right)}$

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

• ${\mathrm{ylog}}_{\mathrm{e}}\left(\mathrm{x}\right) = {\log }_{\mathrm{e}}\left(\mathrm{x}\right) +\hspace{0.17em}1$

• $\mathrm{y} = {\log }_{\mathrm{e}}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{e}$

If u(x) and u(x) are two independent solutions of the differential equation

$\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + \mathrm{b} \frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{cy} = 0,$

then additional solution(s) of the given differential equation is(are)

• y = 5u(x) + 8v(x)

• y = c₁{u(x) - v(x)} + c₂v(x), c₁ and c₂ are arbitrary constants

• y = c₁u(x)v(x) + c₂u(x)v(x), c₁ and c₂ are arbitrary constant

• y = u(x)v(x)

The solution of the differential equation $\left({\mathrm{y}}^2 + 2\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}} = \mathrm{y}$ satisfies x = 1, y = 1. Then, the solution is

• $\mathrm{x} = {\mathrm{y}}^2\left(1 + {\log }_{\mathrm{e}}\left(\mathrm{y}\right)\right)$

• $\mathrm{y} = {\mathrm{x}}^2\left(1 + {\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)$

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

• $\mathrm{y} = {\mathrm{x}}^2\left(1 + {\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)$

A family of curves is such that the length intercepted on the y-axis between the origin and the tangent at a point is three times the ordinate of the point of contact. The family of curves is

• xy = C, C is a constant

• xy² = C, C is a constant

• x²y = C, C is a constant

• x²y² = C, C is a constant

The solution of the differential equation $\mathrm{ysin}\left(\frac{\mathrm{x}}{\mathrm{y}}\right)\mathrm{dx} = \left\{\mathrm{xsin}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) - \mathrm{y}\right\}\mathrm{dy}$ satisfying $\mathrm{y}\left(\frac{\mathrm{\pi }}{4}\right)$ = 1 is

• $\cos \left(\frac{\mathrm{x}}{\mathrm{y}}\right) = {\log }_{\mathrm{e}}\left(\mathrm{y}\right) + \frac{1}{\sqrt{2}}$

• $\sin \left(\frac{\mathrm{x}}{\mathrm{y}}\right) = {\log }_{\mathrm{e}}\left(\mathrm{y}\right) + \frac{1}{\sqrt{2}}$

• $\sin \left(\frac{\mathrm{x}}{\mathrm{y}}\right) = {\log }_{\mathrm{e}}\left(\mathrm{x}\right) - \frac{1}{\sqrt{2}}$

• $\cos \left(\frac{\mathrm{x}}{\mathrm{y}}\right) = - {\log }_{\mathrm{e}}\left(\mathrm{x}\right) - \frac{1}{\sqrt{2}}$

The general solution of the differential equation

$\frac{d\mathrm{y}}{d\mathrm{x}} = \frac{\mathrm{x} +\hspace{0.17em}\mathrm{y} +\hspace{0.17em}1}{2\mathrm{x} +\hspace{0.17em}2\mathrm{y} +\hspace{0.17em}1}$ is

• ${\log }_{\mathrm{e}}\left|3\mathrm{x} + 3\mathrm{y} + 2\right| + 3\mathrm{x} + 6\mathrm{y} = \mathrm{C}$

• ${\log }_{\mathrm{e}}\left|3\mathrm{x} + 3\mathrm{y} + 2\right| - 3\mathrm{x} + 6\mathrm{y} = \mathrm{C}$

• ${\log }_{\mathrm{e}}\left|3\mathrm{x} + 3\mathrm{y} + 2\right| - 3\mathrm{x} - 6\mathrm{y} = \mathrm{C}$

• ${\log }_{\mathrm{e}}\left|3\mathrm{x} + 3\mathrm{y} + 2\right| + 3\mathrm{x} - 6\mathrm{y} = \mathrm{C}$

Let y be the solution of the differential equation

$\mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}} = \frac{{\mathrm{y}}^2}{1 - \mathrm{ylog}\left(\mathrm{x}\right)}$ satisfying y(1) = 1. Then, y satisfies

• y = x^{y - 1}

• y = x^y

• y = x^{y + 1}

• y = x^{y + 2}