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

(b) Order of the differential equation: $ 5x^3 \frac{d^3y}{dx^3} - 3\left(\frac{dy}{dx}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^4 + y = 0 $

Find the shortest distance between the lines: \[ \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}), \] \[ \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}). \]

The differential coefficient of the  \( \sin(x^2 + 5) \) with respect to \( x \) will be:
 

(d) If \( y = x^x \), then find \( \frac{dy}{dx} \).
Find the interval in which the function \[ f(x) = \frac{3}{10}x^4 - \frac{4}{5}x^3 - 3x^2 + \frac{36}{5}x + 11 \] is (A) increasing, (B) decreasing.
(a) If \( x^y = e^{x-y} \), then prove that \[ \frac{dy}{dx} = \frac{\log_e x}{(1 + \log_e x)^2}. \]

Differentiate the  \( \sin mx \) with respect to \( x \).

(e) If \( y = \log_e(\tan x) \), find \( \frac{dy}{dx} \):

If \[ y = 500 e^{7x} + 600 e^{-7x}, \quad \text{then show that} \quad \frac{d^2 y}{dx^2} = 49y. \] 

Solve the differential equation \[ \frac{dy}{dx} + y \cot x = 2x + x^2 \cot x \quad \text{where} \quad x \neq 0. \] 

Solve the differential equation \[ (x - y) \frac{dy}{dx} = x + 2y. \]

Find the intervals in which the function \( f(x) = x^2 - 4x + 6 \) is 

Solve: \[ \frac{dy}{dx} = \frac{1 + y^2}{1 + x^2}. \] 

If \( e^y(x + 1) = 1 \), show that \[ \frac{d^2y}{dx^2} = \left( \frac{dy}{dx} \right)^2. \] 

The order of the differential equation \[ \left( \frac{d^3 y}{dx^3} \right)^2 + x \left( \frac{dy}{dx} \right)^3 + 8y = \log_e x \] is:
Find the maximum value of \( Z = 4x + y \) under the given constraints by graphical method: \[ x + y \leq 50, \quad 3x + y \leq 90, \quad x \geq 0, \quad y \geq 0. \]
Find the general solution of the differential equation \( x \frac{dy}{dx} + 2y = x^2 \); \( x \neq 0 \).
If \( y = x^3 + \tan x \), then find \( \frac{d^2 y}{dx^2} \).
Solve the differential equation \( y \log y \, dx - x \, dy = 0 \).
Find the order of the differential equation \[ \frac{d^3y}{dx^3} + x^2 \left(\frac{d^2y}{dx^2}\right)^3 + \frac{dy}{dx} + y = 0. \]
Find the value of \( \frac{dy}{dx} \) of the curve \( x = t^2 + 3t - 8 \), \( y = 2t^2 - 2t - 5 \) at the point \( (2, -1) \).
Find the particular solution of the differential equation \[ \frac{dy}{dx} + y \cot x = 2x \quad (x \neq 0), \quad \text{when } y = 0 \text{ if } x = \frac{\pi}{2}. \]
Find the area bounded by the circle \[ x^2 + y^2 - 2x = 8. \]
(c) Find the value of \( \int \sqrt{5 + 2x + x^2} \, dx \).
(d) Solve: \( \frac{dy}{dx} = \frac{1 + x}{1 - y} \).