List of top Mathematics Questions on Methods of Solving First Order, First Degree Differential Equations asked in CBSE CLASS XII

If \( \int_0^1 (6x^2 - 4x + k) \, dx = 0 \), then the value of \( k \) is :
The value of \( \int_{-5}^{-1} \frac{1}{x} \, dx \) is equal to :
If \(\int_{0}^{1} \frac{dx}{e^x + e^{-x}} = \tan^{-1} e + k\), then the value of \(k\) is :
Evaluate : \( \int_{0}^{\pi} \frac{\sin^{2026} x}{\sin^{2026} x + \cos^{2026} x} \, dx \)
Find : \( \int \frac{3x - 1}{\sqrt{x^2 - 4x}} dx \)
Evaluate : \( \int_{\pi/12}^{5\pi/12} \frac{dx}{1 + \sqrt{\cot x}} \)
Solve the following differential equation : \( x \frac{dy}{dx} = y - x \sin^2 \left( \frac{y}{x} \right) \), given that y(1) = \( \frac{\pi}{6} \).
Find the general solution of the differential equation : \( y \log y \frac{dx}{dy} + x = \frac{2}{y} \).
The integrating factor of the differential equation \( 2x \frac{dy}{dx} - y = 3 \) is
The integrating factor of differential equation \( R \frac{dx}{dy} + Px = Q \) where \( P, Q, R \) are functions of \( y \) is
Find the general solution of the differential equation \( (x^2 + y^2) \, dy = xy \, dx \).
\( \frac{dy}{dx} = F(x, y) \) will be a homogeneous differential equation for which of the following functions ?
(i) \( F(x, y) = 3x + 2y \)
(ii) \( F(x, y) = \sin \frac{y}{x} + \log y - \log x \)
(iii) \( F(x, y) = e^{y/x} + 1 \)
(iv) \( F(x, y) = \sqrt{x^2 + y^2} - y \)
Solve the following homogeneous differential equation: \( x \frac{dy}{dx} = y - x \sin^2\left(\frac{y}{x}\right) \), given the initial value condition that \( y = \frac{\pi}{6} \) when \( x = 1 \).
The integrating factor of the differential equation $2x \frac{dy}{dx} - y = 3$ is:
The general solution of the differential equation $x \, dy - y \, dx = 0$ is:

The corner points of the feasible region determined by the system of linear constraints are as shown in the following figure: 
(i) If \( Z = 3x - 4y \) be the objective function, then find the maximum value of \( Z \). 
(ii) If \( Z = px + qy \) where \( p, q>0 \) be the objective function, find the condition on \( p \) and \( q \) so that maximum value of \( Z \) occurs at \( B(4, 10) \) and \( C(6, 8) \).