Differential equations practice · Mixed difficulty

Let $y=y(x)$ be the solution of the differential equation $\left(x^{2}-3 y^{2}\right) d x+3 x y \, d y=0, \, y(1)=1$. Then $6 y^{2}(e)$ is equal to

In order to solve the differential equation
$x \cos x \frac{d y}{d x}+y(x \sin x+\cos x)=1$
the integrating factor is:

The solution of $(1+x y) y d x+(1-x y) x d y=0$ is

Solution of $\ell \mathrm{n}(\mathrm{dy} / \mathrm{dx})=4 \mathrm{x}-2 \mathrm{y}-2, \mathrm{y}(1)=1$, is -

The solution of $y' - y = 1,y\left( 0 \right) = - 1$ is given by $y\left( x \right)$, which is equal to

If $\frac{{dy}}{{dx}} + y = 2{e^{2x}}$, then $y$ is equal to

The differential equation of all the lines in the $x y$-plane is

Let $I$ be the purchase value of an equipment and $V(t)$ be the value after it has been used for $t$ years. The value $V(t)$ depreciates at a rate given by the differential equation $\frac{d V}{d t}=-k(T-t)$, where $k>0$ is a constant and $T$ is the total life in years of the equipment. Then the scrap value $V(T)$ of the equipment is:

Let $x = x\left( y \right)$ be the solution of the differential equation $2y{e^{\frac{x}{{{y^2}}}}}dx + \left( {{y^2} - 4x{e^{\frac{x}{{{y^2}}}}}} \right)dy = 0$ such that $x\left( 1 \right) = 0.$ Then $x\left( e \right)$ is equal to

The differential equation of the family of curve ${y^2} = 4a\left( {x + a} \right)$, is

The length of the subnormal of the curve $x^2=4 y$ at point $(-2,1)$ is

The solution of the differential equation $\left( {1 + {y^2}} \right) + \left( {x - {e^{{\mathop\mathrm{t}\nolimits} {{\tan }^{ - 1}}y}}} \right)\frac{{dy}}{{dx}} = 0$ is

If $y = a{x^{n + 1}}$, then ${x^2}\frac{{{d^2}y}}{{d{x^2}}}$ is equal to

$\frac{d^2 x}{d y^2}$ equals to

The solution of $\frac{{dy}}{{dx}} = {\left( {\frac{y}{x}} \right)^{\frac{1}{3}}}$, is

The solution of the equation $d y / d x=\cos (x-y)$, where $y\left(\frac{3 \pi}{2}\right)=\pi$ is

The order of the differential equation of family of curves $y=C_1 \sin ^{-1} x+C_2 \cos ^{-1} x+C_3 \tan ^{-1} x+C_4 \cot ^{-1} x$ (where $C_1, C_2, C_3$ and $C_4$ are arbitrary constants) is

The equation of the curve whose subnormal is equal to a constant $a$ is

The solution of the D.E $y d x-x d y+3 x^2 y^2 d x=0$ is

$\frac{d x}{d y}=\frac{1+x-y^2}{y}$ and $x(1)=1$, then $5 x(2)$ is equal to

The differential equation of all parabola whose axis of symmetry is parallel to $x$-axis is of order

If the differential equation of the family of curve given by $y=A x+B e^{2 x}$, where $A$ and $B$ are arbitrary constants, is of the form $(1-2 x) \frac{d}{d x}\left(\frac{d y}{d x}+l y\right)+k\left(\frac{d y}{d x}+l y\right)=0$, then the ordered pair $(k, l)$ is

If $m$ and $n$ are the order and degree of the differential equation $\left(\frac{d^2 y}{d x^2}\right)^5+4 \frac{\left(\frac{d^2 y}{d x^2}\right)^3}{\frac{d^3 y}{d x^3}}+\frac{d^3 y}{d x^3}=x^2-1,$ then

Order of the differential equation of the family of all concentric circles centred at $\left( {h,k} \right)$ is

The tangent at any point $P$ on the curve $y=f(x)$ meets the $y$-axis in $B . N$ is the foot of the perpendicular drawn from $P$ onto the $x$-axis. If the rectangle with $O B$ and $O N(O$ is the origin) as adjacent sides is of constant area and equals $a^2$, then the equation of the curve is

$y=a e^{-1 / x}+b$ is a solution of $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{y}{x^2}$ when

$f(x)+f^{\prime}(x)=\int\limits_{0}^2 f(t) d t$ and $f(0)=e^{-2}$, then the value of $f(2)-2 f(0)$ is:

The curve whose subtangent is $n$ times the abscissa of the point of contact and passes through $(2,3)$ represent

Which of the following is a second order of differential equation?

The area of the region enclosed between the two circles $x^2+y^2=1$ and $(x-1)^2+y^2=1$, is