Differential Equations

The order and degree of the differential equation $\left(\dfrac{d^3 y}{dx^3}\right)^{\frac{2}{3}} - 3\dfrac{d^2 y}{dx^2} + 5\dfrac{dy}{dx} + 4 = 0$ are respectively

If $\tan x$ is an integrating factor of the differential equation $\dfrac{dy}{dx} + Py = Q$, then $P$ can be

If $y = f(x)$ is a monotonically increasing function such that $\left(\dfrac{dy}{dx}\right)^2 = 6 - \dfrac{dy}{dx}$ and $y(0) = 5$, then $y(3) = \cdots$

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

The general solution of the differential equation $\dfrac{dy}{dx} + \dfrac{y}{x} = x^2 + 5$ is ....

The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{a + bx}{c + dy}$ represents a family of circles centered at the origin if...

The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{x - y}{x + y}$, when $x = 0$ and $y = 0$ represents ....

The solution of the differential equation $\dfrac{dy}{dx} = \cos(x + y)$ is...

If the solution of the differential equation $(1 + x^3)\dfrac{dy}{dx} + 6x^2y = 1 + x^2$ is $y = \dfrac{1}{(1 + x^3)^s}\left[x + \dfrac{x^p}{p} + \dfrac{x^q}{q} + \dfrac{x^r}{r} + c\right]$, then the LCM of $p, q, r$ and $s$ is...

The differential equation of all lines where the length of the normal from the origin is p and the inclination of the normal is $\alpha$ is... (where p and $\alpha$ are arbitrary constants)

The general solution of the differential equation $x\sin x\dfrac{dy}{dx} + (x\cos x + \sin x)y = \sin x$ is

A body cools from $100^\circ\text{C}$ to $60^\circ\text{C}$ in 20 minutes, the temperature of the surroundings being $20^\circ\text{C}$. The total time taken (in minutes) for the body to cool down to $40^\circ\text{C}$ is ...

The equation of the curve whose slope is $\dfrac{y-1}{x^2+x}$ and which passes through the point $(1, 0)$ is

The order and degree of the differential equation $\sqrt{1 + \dfrac{1}{\left(\dfrac{dy}{dx}\right)^2}} = \left(\dfrac{d^2y}{dx^2}\right)^{\frac{3}{2}}$ are $\ldots$

A spherical raindrop evaporates at a rate proportional to its surface area. The differential equation involving the rate of change of its radius $r$ with time '$t$' is $\ldots$ (where $k$ is a positive constant)

The integrating factor of the differential equation $x\dfrac{dy}{dx} + 2y = x^2 \log x$ is

The differential equation of the family of all parabolas whose axis is the $y$-axis is $\ldots$

The rate of disintegration of a radioactive element at any time is proportional to its mass at that time, where $k$ $(k>0)$ is the constant of proportionality. The time during which an original mass of 1.5 gm will disintegrate to a mass of ...

For the differential equation $(x^2 + y^2)\,dy = xy\,dx$, it is given that $y(1) = 1$ and $y(x_0) = e$, then the value of $x_0$ is _____

If $y = e^{-mx}$ is a solution of the differential equation $\dfrac{d^2y}{dx^2} + 4\dfrac{dy}{dx} + 3y = 0$, then the values of $m$ are

The tangent to the curve intersects the Y-axis at point P. A line drawn through point P is perpendicular to this tangent and passes through another point $(1, 0)$. The differential equation of the curve is...

The general solution of the differential equations $\dfrac{dy}{dx} = (9x + y + 5)^2$ is...

The general solution of the differential equation $\sec y + (x - e^{\sin y})\dfrac{dy}{dx} = 0$ is...

The rate of reduction is proportional to the square root of a persons existing assets at that moment. If his assets at the beginning are 10000 and they dwindle down to 5625 in 2 years, then the person will be bankrupt in

The solution of the differential equation $\left(x + 2y^3\right)\dfrac{dy}{dx} - y = 0$ is

The degree of the differential equation obtained from the equation $(y - a)^2 = 4(x - b)$ [where $a$ and b are arbitrary constants] is

The integrating factor of the differential equation $(1 + t^2) + \left(x - e^{\tan^{-1}t}\right)\dfrac{dt}{dx} = 0$ is

If $f(x)$ is a polynomial such that $f(x) = [f'(x)]^2$ and $f(2) = 0$, then $f(-2) = \ldots$

The equation of the curve passing through the point (0, 1) and having a slope of the tangent at point $P(x, y)$ is equal to $\dfrac{y}{x + y}$, is

If water at $100^\circ\text{C}$ cools in 10 minutes to $80^\circ\text{C}$ and to $65^\circ\text{C}$ in the next 10 minutes, then the room temperature will be ...

For the differential equation $\dfrac{d^3y}{dx^3} + \cos\left(\dfrac{d^2y}{dx^2}\right) = 0$, which of the following is true ?

The solution of the differential equation $x^2 \dfrac{dy}{dx} - xy = 1$ is...

If the differential equation $\begin{vmatrix} f(x) & f'(x) \\ f'(x) & f''(x) \end{vmatrix} = 0$ for all $x$ and $f(0) = 1, f'(0) = 2$, then...

The solution of the differential equation $\dfrac{dy}{dx} = e^{x-y} + x^2 e^{-y}$ is...

The solution of the differential equation $e^{-x}(y+1)\text{d}y + (\cos^2 x - \sin 2x)y\, \text{d}x = 0$, given that $y = 1$ when $x = 0$ is

The order and degree of the differential equation $3 - \left(\dfrac{\text{d}^3 y}{\text{d}x^3}\right)^{\frac{7}{3}} = \left(\dfrac{\text{d}y}{\text{d}x}\right)^5$ are respectively

The solution of $\dfrac{\text{d}y}{\text{d}x} = \sin(x + y) + \cos(x + y)$ is

A spherical mothball has initial radius 3 cm. Due to evaporation, the radius of the ball reduces to 1 cm in 4 months. In how many months would the mothball evaporate completely if the volume is lost at a rate proportional to the surface are...

A spherical balloon expands at a rate proportional to its surface area. Initially its radius is 2 cm and 5 minutes later it increases upto 7 cm, then surface area of spherical balloon after 12 minutes will be

The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{x+y}{x-y}$ is

If $\dfrac{dy}{dx} = y + 5$ and $y(0) = 4$ then $y(\log 2)$ is equal to

A body cools according to Newton's law of cooling from $100^\circ$C to $60^\circ$C in $20$ minutes. The temperature of the surroundings being $20^\circ$C, then the total time required for the body to cool down to $30^\circ$C is

The general solution of the differential equation $(x+y)\dfrac{dy}{dx} = 1$ is

The differential equation of $3y = \sqrt[3]{x+c}$ is

The order and degree of the differential equation $\sqrt{2 + \left(\dfrac{d^2 y}{dx^2}\right)^3} = \left(\dfrac{d^3 y}{dx^3}\right)^{5/2}$ respectively are

The differential equation representing the family of curves $x \sin x + y^3 = 4ax$ is

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

The solution of the differential equation $x\dfrac{dy}{dx} + y = x^3 y^6$, is...

The solution of the differential equation $2x\dfrac{dy}{dx} - y = 3$ represents a family of

The integrating factor of the differential equation $x \frac{\mathrm{~d} y}{\mathrm{~d} x}+y \log x=x \cdot \mathrm{e}^x x^{-\frac{1}{2}} \log x(x>0)$ is