The general solution of the differential equation $\left(x-y^2\right) d x+y\left(5 x+y^2\right) d y=0$ is :

The slope of the tangent to a curve $C: y=y(x)$ at any point $[x, y)$ on it is $\frac{2 e^{2 x}-6 e^{-x}+9}{2+9 e^{-2 x}}$. If...

Let $y=y(x), x>1$, be the solution of the differential equation $(x-1) \frac{d y}{d x}+2 x y=\frac{1}{x-1}$, with $y(2)=\frac{1+e}{2 e^4}$. If $y(3) =\frac{\mathrm{e}^\alpha+1}{\beta \mathrm{e}^\alpha}$. Then the...

If $y=y(x)$ is the solution of the differential equation $\left(1+e^{2 x}\right) \frac{d y}{d x}+2\left(1+y^2\right) e^x=0$ and $y(0)=0$, then $6\left(y^{\prime}(0)+\left(y\left(\log _e \sqrt{3}\right)\right)^2\right)$ is equal to:

Let the slope of the tangent to a curve $y=f(x)$ at $(x, y)$ be given by $2 \tan (\cos x-y)$. if the curve passes through...

Let $\mathrm{x}=\mathrm{x}(\mathrm{y})$ be the solution of the differential equation $2 \mathrm{ye}^{\mathrm{x} / \mathrm{y}^2} \mathrm{dx}+\left(\mathrm{y}^2-4 \mathrm{xe}^{\mathrm{x} / \mathrm{y}^2}\right) \mathrm{dy}=0$ such that $\mathrm{x}(1)=0$. Then, $x(e)$ is equal...

The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by $...

If the solution of the differential equation $\frac{d y}{d x}+e^x\left(x^2-2\right) y=\left(x^2-2 x\right)\left(x^2-2\right) e^{2 x}$ satisfies y(0) = 0, then the value of y(2) is _______...