List of top Mathematics Questions

If \[ x^a \frac{dy}{dx} = y^\beta \left( \gamma \log x + \delta y + 1 \right) \] is a homogeneous differential equation, then
The particular solution of the differential equation \[ (1 + y^2) \, dx - xy \, dy = 0, \quad y(1) = 0 \quad \text{represents:} \]
The differential equation \[ y^2 \, dx + (3xy - 1) \, dy = 0 \] is:
If \( y = y(x) \) is the solution of the differential equation \[ x \frac{dy}{dx} = y + xe^{-\frac{y}{x}}, \quad y(1) = \log e \] , then \( y(e) = \)
If $y = y(x)$ is a particular solution of $\sqrt{1 - x^2} \frac{dy}{dx} + \frac{2x}{\sqrt{1 - x^2}} y = x$, $y(0) = 1$, then $y\left(\frac{1}{2}\right) =$
Let \( c_1, c_2, c_3, c_4 \) be arbitrary constants. The order of the differential equation corresponding to \[ y = c_1 e^x + c_2 e^{\log_e x} + c_3 \sin^2 x - c_4 (\cos^5 x - 1) \] is:
If $X=x+h, Y=y+k$ transforms $\frac{dy}{dx} = \frac{2x+3y-7}{3x+2y-8}$ to a homogeneous differential equation, then $(h,k)=$
Let \( f(x) \) be a differentiable function such that \( f(1) = 2 \), \( f(2) = 6 \), and \[ f(x+y) = f(x) + kxy + \frac{4}{3} y^2, \quad \forall x, y \in \mathbb{R}. \] Then \( f(x) = \) ?
If a curve passes through (1, 2) and has the slope of its tangent \(1 - \frac{1}{x^2}\) at a point \((x, y)\), then the equation of that curve is
a, b, c, d are arbitrary constants. Then the corresponding differential equation to \( y = ae^x + be^{-x} + c \cos x + d \sin x \) is
Let \[ f(x) = \begin{cases} \frac{5e^{|x|} + 2}{3 - e^{|x|}}, & x \ne 0 \\ 0, & x = 0 \end{cases} \] Then at \( x = 0 \), the function \( f(x) \) is:
The general solution of \( \frac{dy}{dx} = \cos^2(x - y - 1) \) is given by \( x = \):
The differential equation having \( y = (a + b)e^{cx+d} \) as its general solution, where \( a, b, c, d \) are arbitrary constants, is
Which one of the following is a homogeneous differential equation?
If \( (3 + 2\sqrt{2})^{x^2 - 4} + (3 - 2\sqrt{2})^{x^2 - 4} = 6 \), then \( x^4 + x^2 + 5 = \)?
If \( y = x \log \left( \frac{1}{ax} \right) \), then \( x(1 + x) \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - y = \)
If \( \lim_{x \to \infty} y(x) = \frac{\pi}{2} \), then the solution of \( x^3 \sin y \frac{dy}{dx} = 2 \) is \( \cos y = \)
The locus of the variable point \( z = x + iy \) whose amplitude is always equal to \( \theta \), is:
If \( \alpha \) is the real root and \( \beta \), \( \gamma \) are the other roots of the equation \( x^3 - a^3 = 0 \, (a>0) \), then the number of common points of the curves given by \( |z - \beta| = \frac{\sqrt{3a}}{2} \) and \( |z - \gamma| = \frac{\sqrt{3a}}{2} \) is:
The product of the four values of \( \left( 1 + i\sqrt{3} \right)^{3/4} \) is:
Arg \[ \frac{4 + 2i}{1 - 2i} + \frac{3 + 4i}{2 + 3i} \] lies in the interval \[ \text{Arg} \left( \frac{4 + 2i}{1 - 2i} + \frac{3 + 4i}{2 + 3i} \right) \]
If $ 1, \omega, \omega^2 $ are the cube roots of unity, then the roots of the equation $ 8z^3 - 12z^2 + 6z - 28 = 0 $ are:
The modulus of the conjugate of \( z = \frac{-2 + i}{(1 - 2i)^2} \) is:
The number of all possible solutions of the equation \( z^3 + z = 0 \) is:
If \( 1, \omega, \omega^2 \) are the cube roots of unity and \( f(x, y) = (x + y)(x\omega + y\omega^2)(x\omega^2 + y\omega) \), then \( f(2, 3) \) is: