List of top Mathematics Questions on Differential Equations

Let \( y = y(x) \) be the solution of the differential equation: 
\[ \frac{dy}{dx} + \left( \frac{6x^2 + (3x^2 + 2x^3 + 4)e^{-2x}}{(x^3 + 2)(2 + e^{-2x})} \right)y = 2 + e^{-2x}, \quad x \in (-1, 2) \] 
satisfying \( y(0) = \frac{3}{2} \). 
If \( y(1) = \alpha \left(2 + e^{-2}\right) \), then the value of \( \alpha \) is ________.

Let \(y = y(x)\) be the solution of the differential equation \(\frac{dy}{dx} = (1 + x + x^2)(1 - y + y^2)\), \(y(0) = \frac{1}{2}\). Then \((2y(1) - 1)\) is equal to:
Let \( x = x(y) \) be the solution of the differential equation} \[ 2y^2 \frac{dx}{dy} - 2xy + x^2 = 0, \quad y>1, \quad x(e) = e. \] Then \( x(e^2) \) is equal to:
If the curve \(y=f(x)\) passes through the point \((1,e)\) and satisfies the differential equation} \[ dy=y(2+\log_e x)\,dx,\quad x>0, \] then \(f(e)\) is equal to:
Let \(y=y(x)\) be the solution curve of the differential equation} \[ (1+\sin x)\frac{dy}{dx}+(y+1)\cos x=0,\qquad y(0)=0. \] If the curve passes through the point \( \left(\alpha,-\frac12\right) \), then a value of \( \alpha \) is:
If the degree of the differential equation $\left(\frac{d^{2}y}{dx^{2}}\right)^{3/2}+5\left(\frac{d^{2}y}{dx^{2}}\right)^{5/2}=7y$ is $m$ and its order is $n$, then $y=Ae^{mx}+Be^{nx}$ is solution of the differential equation
If $y=f(x)$ is a solution of $\frac{dy}{dx}=(y-kx)^{2}+K$ when $x=0$ and $y=1$, then $f(2)=$
At a point $P(x,y)$ on a curve $x=f(y)$, the x-intercept of the tangent is always equal to the y-coordinate of the point of contact, then $f(y)=$
Consider the differential equations \(\frac{dy}{dx}(x+y+1)=1\) and \(\frac{dx}{dy}=3y+2x^2\).
Which of the following is correct regarding these two differential equations?
Among the following differential equations, the equation having order 2 and degree 3 is:
The general solution of the differential equation \( \frac{dy}{dx} + y \tan x = - \tan x \log(\cos x) \) (\(0 < x < \pi/2\)) is:
If the differential equation of the family of curves given by \[ y=ae^x+b\cos x, \] where \(a\) and \(b\) are arbitrary constants, is \[ y_2(\cos x+\sin x)+y(\cos x-\sin x)=2y_1f(x), \] then \(f(x)=\)
The general solution of the differential equation \[ (4xy^2-2xy+2x^2y^2-x^2y)\,dx=dy \] is
The differential equation among the following, whose general solution is \( y=A e^{5x}+B e^{-4x} \) is
Solve $(1+y^2)+(x-e^{-\tan^{-1}y})\frac{dy}{dx}=0$
The general solution of the differential equation \[ (x+y-1)\,dy=(x-y+1)\,dx \] is
The substitution required to reduce the differential equation \( \frac{dy}{dx}+\sin y \cos y \sin x = \sin 2x \cos^{2}y \) to a linear differential equation in z is
The integrating factor of the differential equation $(1 + x^2) \frac{dy}{dx} + 2xy = \cos x$ is:
The solution of the differential equation $x \frac{dy}{dx} + 2y = x^2$ is:
Solve $\cos(x+y)dy=dx$
The general solution of the differential equation \( \frac{dy}{dx}=x^{2}y(x+y+xy+1) \) is \( y = \)
Eliminate constants from $y=A(x+B)^2$
The order and degree of the differential equation $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2} = \frac{d^2y}{dx^2}$ are respectively:
The general solution of the differential equation \[ \frac{dy}{dx}=\frac{2xy-3y^2}{2x^2+3xy} \] is
Which of the following equations is NOT a Linear Differential Equation?