List of top Mathematics Questions on Differential Calculus

If $ y = f(x) $ satisfies the differential equation \[ (x^2 - 4)y' - 2xy + 2x(4 - x^2)^2 = 0 \] and $ f(3) = 15 $, then find the local maximum value of $ f(x) $:

Let $ f : \mathbb{R} \to \mathbb{R} $ be a twice differentiable function such that \[ (\sin x \cos y)(f(2x + 2y) - f(2x - 2y)) = (\cos x \sin y)(f(2x + 2y) + f(2x - 2y)), \] for all $ x, y \in \mathbb{R}. $

If $ f'(0) = \frac{1}{2} $, then the value of $ 24f''\left( \frac{5\pi}{3} \right) $ is:

If y = 3e2x + 2e3x, then $\frac{d^2y}{dx^2} + 6y$ is equal to

If the straight line $ \frac{x}{p} - \frac{y}{q} = 1 $ touches the curve $ \left( \frac{x}{a} \right)^n - \left( \frac{y}{b} \right)^n = 1 $, then

If y = 3e2x + 2e3x, then $\frac{d^2y}{dx^2} + 6y$ is equal to

If the function $ f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 $, where $ a>0 $, attains its local maximum and minimum at $ p $ and $ q $, respectively, such that $ p^2 = q $, then $ f(3) $ is equal to:

A cylindrical tank of radius 10 cm is being filled with sugar at the rate of 100π cm3/s. The rate at which the height of the sugar inside the tank is increasing is:

Let $ f(x) = |x - \alpha| + |x - \beta| $, where $ \alpha, \beta $ are the roots of the equation $ x^2 - 3x + 2 = 0 $. Then the number of points in $ [\alpha, \beta] $ at which $ f $ is not differentiable is:

If $f(x) = \begin{cases} x^2 + 3x + a, & x \leq 1 bx + 2, & x>1 \end{cases}$, $x \in \mathbb{R}$, is everywhere differentiable, then

The value of the integral $ \int_{0}^{\pi/2} \log\left(\frac{4 + 3\sin x}{4 + 3\cos x}\right) dx $ is:

The equation of the tangent to the curve x^5/2+ y^5/2 = 33 at the point(1, 4) is:

If $ e^y = x^x $, then which of the following is true?

If $y(\theta) = \frac{2\cos\theta + \cos2\theta}{\cos3\theta + 4\cos2\theta + 5\cos\theta + 2}$, then at $\theta = \frac{\pi}{2}, y'' + y' + y$ is equal to:

Find the general solution of the differential equation $ y \, dx - x \, dy + (x \log x) \, dx = 0 $.

The number of arbitrary constants in the general solution of the differential equation dy/dx + y = 0 is:

Check whether the function $ f(x) = x^2 |x| $ is differentiable at $ x = 0 $ or not.
Answer: $ f(x) $ is differentiable at $ x = 0 $.

If $ y = 5\cos x - 3\sin x $, then $ \frac{d^2y}{dx^2} + y $ equals:

When $ y = vx $, the differential equation \[ \frac{dy}{dx} = \frac{y}{x} + \frac{f\left( \frac{y}{x} \right)}{f'\left( \frac{y}{x} \right)} \] reduces to:

Let $ V = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 \mid x_1 = x_2 \} $. Consider $ V $ as a subspace of $\mathbb{R}^4$ over the real field. Then the dimension of $ V $ is equal to __________ (answer in integer).