Variable Separable Form

Let the tangent at any point $\mathrm{P}$ on a curve passing through the points $\left(1,1\right)$ and $\left(\frac{1}{10},100\right)$, intersect positive $\mathrm{x}$-axis and $\mathrm{y}$-axis at the points $\mathrm{A}$ and $\mathrm{B}$ respectively. If $P A: P B=1: k$ and $\mathrm{y}=\mathrm{y}\left(\mathrm{x}\right)$ is the solution of the differential equation $e^{\frac{dy}{dx}}=kx+\frac{k}{2},y\left(0\right)=k$, then $4\mathrm{y}\left(1\right)-5{\log }_{\mathrm{e}}3$ is equal to _______________

Let $y=y_1\left(x\right)$ and $y=y_2\left(x\right)$ be the solution curves the differential equation $\frac{dy}{dx}=y+7$ with initial conditions $y_1\left(0\right)=0$ and $y_2\left(0\right)=1$ respectively. Then the curves $y=y_1\left(x\right)$ and $y = y_2\left(x\right)$ intersect at

Let $y=y\left(x\right)$ be a solution curve of the differential equation, $\left(1-x^2{y}^2\right)dx=ydx+xdy$, If the line $x=1$ intersects the curve $y=y\left(x\right)$ at $y=2$ and the line $x=2$ intersects the curve $y=y\left(x\right)$ at $y=\alpha$, then a value of $\alpha$ is

Let a curve $y=f\left(x\right), x\in \left(0, \infty \right)$ pass through the points $P\left(1, \frac{3}{2}\right)$ and $Q\left(a, \frac{1}{2}\right)$. If the tangent at any point $R(b, f(b\left)\right)$ to the given curve cuts the $y$-axis at the point $S(0, c)$ such that $bc=3,$ then ${\left(PQ\right)}^2$ is equal to _____.

If $\frac{dy}{dx}=6e^x+e^{2x}+e^{3x}$, then $y\left(2\right)-y\left(0\right)$ is

If $\frac{dy}{dx}=y+7$ and $y\left(0\right)=0$, then the value of $y\left(1\right)$ is

If solution of differential equation $\left(1-x^2{y}^2\right)dx=xdy+ydx$ is $y\left(x\right)$ and $y\left(2\right)=4$, then $\frac{5y\left(5\right)+1}{5y\left(5\right)-1}$ is

Given $y\left(0\right)=200$ and $\frac{dy}{dx}=32000-20y^2$, then find the value of $\underset{x\to \infty }{\lim }y\left(x\right)$.

The solution of the differential equation $\frac{dy}{dx}-\frac{\left(y+3x\right)}{{\log }_e\left(y+3x\right)}+3=0$ is:

(where, $C$ is a constant of integration)

Solution of $\left(1+e^x\right)y{y}^{\text{'}}=e^x$ when $y\left(0\right)=1$ is

The solution of the differential equation $\frac{dy}{dx}-\frac{y+3x}{{\log }_e\left(y+3x\right)}+3=0$, where $c$ is constant of integration

The solution of the differential equation $\left(y^2+x{y}^2\right)\frac{dx}{dy}+x^2-y{x}^2=0$ is

At any point on the curve $y=f\left(x\right)$ the length of the sub-normal is constant, then the curve is

Let $p\left(x\right)$ be a polynomial of least degree having maximum value $6$ at $x=1$ and minimum value $2$ at $x=3$, then $p\left(2\right)+p^{\text{'}}\left(0\right)=$

The general solution of the differential equation $ydy+\sqrt{1+y^2}dx=0$ represents a family of

If the derivative of $f\left(x\right)$ w.r.t $x$ is $\frac{\cos 2x}{2f\left(x\right)}$, then period of $f\left(\frac{\mathrm{\pi }x}{3}\right)$ is-

A curve $P$ passing through $\left(1, 1\right)$ and whose slope of the tangent at a point $\left(x, y\right)$ is $-\frac{x}{y}$. If $L_1$ and $L_2$ are the length of tangents drawn from $\left(\sqrt{6}, 0\right)$ on the curve $P$, then the value of $\frac{L_1+L_2}{2}$ is equal to

The solution of the differential equation $\frac{dy}{dt}-\sin t=e^{2t}+t^2$, when $y\left(0\right)=1$ is

Let differential equation $\frac{dy}{dx}=\frac{x\mathrm{sec\theta }+6}{2y\mathrm{tan\theta }+3}$ represents a real circle, then ${\sin }^2\mathrm{\theta }+{\mathrm{cosec}}^2\mathrm{\theta }$ is ($\mathrm{\theta }$ is a real constant) 

Let $f:R\to R$ be a differentiable fuction such that $f\left(\frac{x+2y}{4}\right)=\frac{f\left(x\right)+3f\left(y\right)}{4},$ where $f\left(0\right)=0,f^{\text{'}}\left(0\right)=2,$ then which of the following is correct.