Equations Reducible to Standard Forms

Q. The solution lof $\frac{dy}{dx}-xtan(y-x)=1$

1. $ce\frac{e^{x^2}}{2}=sin(y-x)$
2. $ce\frac{e^{x^2}}{2}=sin(y+x)$
3. $ce\frac{e^{x^2}}{2}=sin(y-x{)}^2$
4. $ce\frac{e^{x^2}}{2}=cos(y-x)$

Q. The general solution of the differential equation $(y^2+e^{2x})dy-y^{3}dx=0$ (C being the constant of integration), is

1. $y^2{e}^{-2x}+2lny=c$
2. $y^2{e}^{-2x}-2lny=c$
3. $y^2{e}^{-2x}-\frac{1}{2}ylny=c$
4. $y^2{e}^{-2x}-\frac{1}{2}lny=c$

Q. The general solution of the differential equation $(y^2+e^{2x})dy-y^{3}dx=0$ (C being the constant of integration), is

1. $y^2{e}^{-2x}+2lny=c$
2. $y^2{e}^{-2x}-2lny=c$
3. $y^2{e}^{-2x}-\frac{1}{2}ylny=c$
4. $y^2{e}^{-2x}-\frac{1}{2}lny=c$

Q. The solution lof $\frac{dy}{dx}-xtan(y-x)=1$

1. $ce\frac{e^{x^2}}{2}=sin(y-x)$
2. $ce\frac{e^{x^2}}{2}=sin(y+x)$
3. $ce\frac{e^{x^2}}{2}=sin(y-x{)}^2$
4. $ce\frac{e^{x^2}}{2}=cos(y-x)$

Q.

Let f(x) is a continuous function which takes positive values for x (x>0), and satisfy ${\int }_0^{x}f\left(t\right)dt=x\sqrt{f\left(x\right)}$ with $f\left(1\right)=\frac{1}{2}$. Then the value of $f(\sqrt{2}+1)$ equals

1. 1
2. $\sqrt{2}-1$
3. $\frac{1}{4}$
4. $\frac{1}{\sqrt{2}-1}$

Q. The slope of the tangent at (x, y) to a curve passing through $(1,\frac{\pi }{4})$ is given by $\frac{y}{x}-co{s}^2\left(\frac{y}{x}\right)$, then the equation of the curve is

1. $y=ta{n}^{-1}\left[log\left(\frac{e}{x}\right)\right]$
   
2. $y=xta{n}^{-1}\left[log\left(\frac{x}{e}\right)\right]$
   
3. $y=xta{n}^{-1}\left[log\left(\frac{e}{x}\right)\right]$
   
4. $Noneofthese$

Q. A solution of $y=2x\left(\frac{dy}{dx}\right)+x^2(\frac{dy}{dx}{)}^4$ is

1. $y=2c^{\frac{1}{2}}{x}^{\frac{1}{4}}+c$
2. $y=2\sqrt{c{x}^2}+c^2$
3. $y=2\sqrt{c}(x+1)$
4. $y=2\sqrt{cx}+c^2$

Q. $\mathrm{Is}\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{x+y+1}{2x+2y+3}\mathrm{reducible}\mathrm{to}\mathrm{variable}\mathrm{separable}\mathrm{form}?$

1. False
2. True

Q. The slope of the tangent at (x, y) to a curve passing through $(1,\frac{\pi }{4})$ is given by $\frac{y}{x}-co{s}^2\left(\frac{y}{x}\right)$, then the equation of the curve is

1. $y=ta{n}^{-1}\left[log\left(\frac{e}{x}\right)\right]$
   
2. $y=xta{n}^{-1}\left[log\left(\frac{x}{e}\right)\right]$
   
3. $y=xta{n}^{-1}\left[log\left(\frac{e}{x}\right)\right]$
   
4. $Noneofthese$

Q. The solution of $\frac{dy}{dx}+1=e^{x+y}$ is

1. $e^{-(x+y)}+x+c=0$
2. $e^{-(x+y)}-x+c=0$
3. $e^{x+y}+x+c=0$
4. $e^{x+y}-x+c=0$

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

1. y =2
2. y =2x
3. y =2x -4
4. $y=2x^2-4$

Q. $\mathrm{The}\mathrm{solution}\mathrm{of}\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{x+y+1}{2x+2y+3}\mathrm{is}$.

1. $\frac{2}{3}\left(x+y\right)+\frac{1}{9}\log \left(2x+2y+4\right|=x+c$
2. $\frac{2}{3}\left(x+y\right)+\frac{1}{9}\log \left(3x+3y+4\right|=x+c$
3. $\frac{2}{3}\left(x+y\right)+\frac{1}{9}\log \left(3x+3y\right|=x+c$
4. $\frac{1}{9}\log \left(3x+3y+4\right|=x+c$