List of top Mathematics Questions on Differential Equations

If $y = f(x)$ is the solution of the differential equation $(1 + \sin x) \frac{dy}{dx} + \cos x = 0$, such that $f(0) = 0$, then $f\left(\frac{\pi}{2}\right)$ is:

Let \( f(x) = \begin{cases} \frac{ax^2 + 2ax + 3}{4x^2 + 4x - 3}, & x \neq -\frac{3}{2}, \frac{1}{2} \\ b, & x = -\frac{3}{2}, \frac{1}{2} \end{cases} \) be continuous at \( x = -\frac{3}{2} \). If \( f(x) = \frac{7}{5} \), then \( x \) is equal to :

Let \( y = y(x) \) be the solution of the differential equation \[ x\frac{dy}{dx} - \sin 2y = x^3(2 - x^3)\cos^2 y,\; x \ne 0. \] If \( y(2) = 0 \), then \( \tan(y(1)) \) is equal to:

Let \( f : [1,\infty) \to \mathbb{R} \) be a differentiable function. If
\[ 6\int_{1}^{x} f(t)\,dt = 3x f(x) + x^3 - 4 \] for all \( x \ge 1 \), then the value of \( f(2) - f(3) \) is

Let $y=y(x)$ be a differentiable function in the interval $(0,\infty)$ such that $y(1)=2$, and \[ \lim_{t\to x}\left(\frac{t^2y(x)-x^2y(t)}{x-t}\right)=3 \text{ for each } x>0. \] Then $2y(2)$ is equal to

Let the solution curve of the differential equation \[ x\,dy - y\,dx = \sqrt{x^2+y^2}\,dx,\quad x>0, \] with $y(1)=0$, be $y=y(x)$. Then $y(3)$ is equal to

If \[ \sec x \frac{dy}{dx} - 2y = 2 + 3\sin x \] and \[ y(0) = -\frac{7}{4}, \] then \( y\left( \frac{\pi}{6} \right) \) is:

If \( y = y(x) \) and \( (1 + x^2) \frac{dy{dx} + (1 - \tan^{-1}x)dx = 0 \) and \( y(0) = 1 \), then \( y(1) \) is}

The solution of the differential equation \[ x\,dy - y\,dx = \sqrt{x^2 + y^2}\,dx \] (where \(c\) is the constant of integration) is

Let $f(x)$ be a differentiable function satisfying the equations $\lim_{t \to x} \dfrac{t^2 f(x)-x^2 f(t)}{t-x} = 3$ and $f(1)=2$. Find the value of $2f(2)$.

If $y=y(x)$ satisfies the differential equation \[ 16(\sqrt{x}+9\sqrt{x})(4+\sqrt{9+\sqrt{x}})\cos y\,dy=(1+2\sin y)\,dx,\quad x>0 \] and \[ y(256)=\frac{\pi}{2},\quad y(49)=\alpha, \] then $2\sin\alpha$ is equal to

Find the general solution of the differential equation \[ y\log y\,\frac{dx}{dy}+x=\frac{2}{y}. \]

Solve the differential equation \[ x\frac{dy}{dx}=y-x\sin^2\left(\frac{y}{x}\right), \quad \text{given that } y(1)=\frac{\pi}{6}. \]

The integrating factor of the differential equation $2x\frac{dy}{dx}-y=3$ is

The general solution of the differential equation $xdy-ydx=0$ is

Find the general solution of differential equation \( ydx + (x - y^2)dy = 0 \).

Find the degree of the differential equation \[xy \frac{d^2y}{dx^2} + x \left( \frac{dy}{dx} \right)^2 - y \left( \frac{dy}{dx} \right) = 2\]

The degree of differential equation \[ 9 \frac{d^2y}{dx^2} = \left\{ 1 + \left( \frac{dy}{dx} \right)^2 \right\}^{\frac{1}{3}} \text{ is} \]

Let $y = y(x)$ be the solution of the differential equation $\sec x \frac{dy}{dx} - 2y = 2 + 3\sin x, x \in (-\frac{\pi}{2}, \frac{\pi}{2})$. If $y(0) = -\frac{7}{4}$, then $y(\frac{\pi}{6})$ is equal to :

Let \(y=y(x)\) be the solution curve of the differential equation \((1+x^2)dy+(y-\tan^{-1}x) dx=0\), \(y(0) = 1\). Then the value of \(y(1)\) is:

If the solution curve $y = f(x)$ of the differential equation $(x^2 - 4) y' - 2xy + 2x(4 - x^2)^2 = 0, x>2$, passes through the point $(3, 15)$, then the local maximum value of $f$ is ___