List of top Mathematics Questions on Differential equations asked in UP Board XII

(e) Find the slope of the curve \( y = 2x^2 - 3\cos x \) at \( x = 0 \).
(b) Solve the differential equation \( \sec^2 x \tan y \, dx + \sec^2 y \tan x \, dy = 0 \).
(a) Find the slope of the curve \( ay^2 = x^3 \) at the point \( (am^2, am^3) \).
(a) Differential coefficient of \( \sin^{-1}(e^{-x}) \) will be:

Find the area of the bounded region of

\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]

The order of the differential equation \( \left( \frac{d^3 y}{dx^3} \right)^2 + x \left( \frac{dy}{dx} \right)^3 + 8y = \log x \) is:
Solve: \( \frac{dy}{dx} = -4xy^2 \).
Solve the differential equation \( \frac{dy}{dx} + y = 1 \) (\( y \neq 1 \)).
Solve: \( y \, dx - (x + 2y^2) \, dy = 0 \).
Solve: \( \frac{dy}{dx} = \frac{x^2 + y^2}{2xy} \).
If \( y = e^{a \cos^{-1} x}, -1 \leq x \leq 1 \), then prove that \( (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} - a^2 y = 0 \).
The order of the differential equation $2x^2 \frac{d^2 y}{dx^2} - 3 \frac{dy}{dx} + y = 0$ will be:
Find the degree of the differential equation \[ x y \left( \frac{d^2 y}{dx^2} \right)^2 + x \frac{dy}{dx} - y = 2. \]
If \( y = A \sin x + B \cos x \), then find the differential equation of it.
The nature of the differential equation \[ (x - y) \frac{dy}{dx} = x + 2y \] will be:
Eliminating arbitrary constants $a$ and $b$, find the differential equation represented by the family of curves: \[ y^2 = a(b^2 - x^2) \]
If $y = e^{a \cos^{-1}x}, \; -1 \leq x \leq 1$, then prove that \[ (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} - a^2 y = 0. \]
The order of the differential equation $\left(\dfrac{d^{3}y}{dx^{3}}\right)^{2} + x^{2}\left(\dfrac{d^{2}y}{dx^{2}}\right)^{3} + 7y = \sin x$ will be:
Solve the differential equation: \[ \frac{dy}{dx} - y = \cos x \]
Solve the differential equation $(\tan^{-1}y - x)dy = (1+y^{2})dx$.
If $y = A\cos\theta + B\sin\theta$, then prove that $\dfrac{d^{2}y}{d\theta^{2}} = -y$.
Solve the differential equation $\log\left(\dfrac{dy}{dx}\right) = 3x + 4y$.
Solve $\dfrac{dy}{dx} = \dfrac{x + e^x}{y}$.