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

Show that \( y = c_1 e^{ax} \cos(bx) + c_2 e^{ax} \sin(bx) \), where \( c_1, c_2 \) are constants, is a solution of the differential equation \( \frac{d^2y}{dx^2} - 2a\frac{dy}{dx} + (a^2 + b^2)y = 0 \).

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

Find the differential coefficient of \(y = x^{\cos x} + (\sin x)^x\) with respect to x.

If \( y = A \cos \theta + B \sin \theta \), then prove that \( \frac{d^2y}{d\theta^2} = -y \).

Solve the differential equation \( \frac{dy}{dx} = e^x \cos x \).

If \( y = A + Be^x \), then prove that \( \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0 \), where A and B are constants.

Solve the differential equation \( \frac{dy}{dx} = \frac{x^2 - 1}{y^2 + 1} \).

Find the particular solution of the differential equation \( \frac{dy}{dx} + y \cot x = 2x + x^2 \cot x, (x \neq 0) \). It is given that \( y = 0 \) if \( x = \frac{\pi}{2} \).

Solve the differential equation \( ydx - (x + 2y^2) dy = 0 \).

The order of the differential equation \( \frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} + y \cos x = 0 \) is :

Find the order and degree of the differential equation \( \frac{d^2y}{dx^2} + \frac{dy}{dx} + y \cdot \sin x = 0 \).

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

Find the differential equation of the family of curves \[ y = a \sin(x + b), \] where \( a \) and \( b \) are arbitrary constants.

Solve the differential equation: \[ \tan^{-1}(y - x) \, dy = (1 + y^2) \, dx. \]

Solve the differential equation \( (x - y) \, dy - (x + y) \, dx = 0 \).

The degree of differential equation \[ \frac{d^2y}{dx^2} = \left( y + \frac{dy}{dx} \right)^{\frac{1}{5}} \] will be