List of practice Questions

The function \[ x+y=\tan^{-1}y \] is the solution of which of the following differential equations?

If $\dfrac{dy}{dx}=y\tan x$, then the solution of the differential equation is:

If the tangent at any point $(x,y)$ of a curve intercepts equal lengths on the coordinate axes, then the differential equation of the curve is:

Let the population of a species of birds surviving at a time \(t\) be governed by the differential equation \[ \frac{dp}{dt}-p=-100 \] If \(p(0)=50\), then \(p(-\ln2)\) is equal to:

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

Find the integrating factor (I.F.) for the differential equation \[ \frac{dy}{dx} + y\sec x = \tan x . \]

Integrating factor of the differential equation $(1 - x^2)\frac{dy}{dx} - xy = 1$ is

$\int x f(x) dx + \frac{f(x)}{2} = 0$, then $f(x)$ is equal to

Sum of the squares of the order and degree (if defined) of a differential equation $2y' + (y'')^2 = \sqrt{y'' - 3}$ is

The solution of the differential equation $(y+x^{2})dx=xdy, x>0$ is a curve which passes through the point (1,0). The equation of the curve is

Elimination of arbitrary constants $A$ and $B$ from $y = Ae^x + Be^{-2x}$ gives the differential equation:

The solution of the differential equation $(x + 2y)dx + (2x - y)dy = 0$ is

Find the general solution of the differential equation \(\dfrac{dy}{dx} + y\cot x = \csc x\).

Calculate \( y(\log 2) \) if \( \dfrac{dy}{dx} = y + 5 \) and \( y(0) = 4 \).

Solve the differential equation \( \frac{dy}{dx} = \frac{1+y^2}{1+x^2} \) given \(y(0)=1\).

Find the general solution of the differential equation \( \frac{dy}{dx} + y = e^{-x} \).

Find the general solution of the differential equation \( \dfrac{dy}{dx} + \dfrac{y}{x} = x^2 \).

If \((x\sqrt{1-x^2}) \, dy - (y\sqrt{1-x^2} - x^2 \cos^{-1} x) \, dx = 0\) and \(\lim_{x \to 1^-} y(x) = 1\), then \(y\left(\frac{1}{2}\right)\) is

Let $y = y(x)$ be the solution of the differential equation $(x^2 - x\sqrt{x^2 - 1}) dy + (y(x - \sqrt{x^2 - 1}) - x) dx = 0, x \geq 1$. If $y(1) = 1$, then the greatest integer less than $y(\sqrt{5})$ is ________.

The number of arbitrary constants in the particular solution of a differential equation of third order are _____

The order and the degree of the differential equation \[ \sqrt{1 + \left(\frac{d^2y}{dx^2}\right)^2} = \sqrt{x + \left(\frac{dy}{dx}\right)^6} \] are respectively _____ and _____

The differential equation of all circles of radius a is of order

Solution of the differential equation Xdy - Ydx = 0 represents

Solve the differential equation:
\[ y \, dx + (x - y^2) \, dy = 0 \]

Solve xy dydx = e^x