List of top Mathematics Questions on General and Particular Solutions of a Differential Equation

If y = f(x) is solution of differential equation \((x^2 - 1) dy = ((x^3 + 1) + \sqrt{(1 - x^2)}dx \) and \(y(0) = 2\) then find \(y(\frac{1}2)\)

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=\frac{4 y^3+2 y x^2}{3 x y^2+x^3}, y(1)=1$ If for some $n \in N , y (2) \in[ n -1, n )$, then $n$ is equal to ______

Let y = y(x) be a solution of the differential equation (x cos x)dy +(xy sin x + y cos x – l)dx = 0, 0 < x < \(\frac{π}{2}\). If\(\frac{π}{3}y(\frac{π}{3})=\sqrt3\), then \(| \frac{π}{6 }y“( \frac{π}{6})+2 y'( \frac{π}{6})| \) is equal to______.

If $y=y(x)$ is the solution curve of the differential equation $\frac{d y}{d x}+y \tan x=x \sec x, 0 \leq x \leq \frac{\pi}{3}, y(0)=1$ then $y\left(\frac{\pi}{6}\right)$ is equal to

The number of solutions of the equation \(cos \bigg(x+\frac{π}{3}\bigg)cos\bigg(\frac{π}{3-x}\bigg)=\frac{1}{4}cos^22x,x ∈ [-3π, 3π]\) is:

The number of solutions of $\frac{d y}{d x}=\frac{y+1}{x-1}$ when $y(1)=2$ is

The integral $ \int \frac{2x^{3} -1}{x^{4} +x} dx $ is equal to : (Here C is a constant of integration)

The integral $\int sec^{2/3} x cosec^{4/3}x\, dx$ is equal to (Hence $C$ is a constant of integration)

The integral $\int \cos( \log \; x)dx$ is equal to : (where C is a constant of integration)

If $\sqrt{1-x^{6} }+ \sqrt{1-y^{6}} =a\left(x^{3} -y^{3}\right) , $ then $ y^{2} \frac{dy}{dx} = $

The equation of the curve passing through the point $\left(a, -\frac{1}{a}\right)$ and satisfying the differential equation $y-x \frac{dy}{dx}=a\left(y^{2}+\frac{dy}{dx}\right)$ is

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 differential equation representing the family of ellipses having foci either on the x-axis or on the y-axis, centre at the origin and passing through the point $(0, 3)$ is :

The curve satisfying the differential equation, $(x^2 - y^2) dx + 2xydy = 0$ and passing through the point $(1, 1)$ is :

The solution of the differential equation $\frac{y dx + x dy}{y dx - x dy} = \frac{x^{2} e^{xy}}{y^{4}} $ satisfying $y(0) = 1$, is :

The number of solutions of $\sin \, x + \sin \, 3x + \sin \, 5x = 0$ in the interval $\left[\frac{\pi}{2} , 3 \frac{\pi}{2}\right] $ is

The number of solutions of $\sin \, 3x = \cos \, 2x$, in the interval $\left( \frac{\pi}{2} , \pi \right)$ is :

A differential equation $ \frac{dy}{dx}= \left(y+4x -4\right)^{2} $ is given. At $ x = 0, y = 4 $ the solution of this differential equation is given by

The integrating factor of the differential equation $x . \frac{dy}{x} + 2y = x^2 $ is $(x \neq 0)$

The integral $\displaystyle\int \frac{dx}{(1+ \sqrt{x}) \sqrt{x - x^2}}$ is equal to (where $C$ is a constant of integration)

The differential equation of the family of circles touching $y$-axis at the origin is

The Solution of the differential equation $\left(x+2y^{3}\right) \frac{dy}{dx}=y$ is

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

The differential equation of the family of parabolas $y^2 = 4ax$, where $a$ is parameter, is

The value of \(\lambda\), such that the following system of equations \(2x - y - 2z = 2\); \(x - 2y + z = -4\); \(x + y + \lambda z = 4\) has no solution, is