Variable Separable Method

Q.

If a $functionF$ is such that $F\left(0\right)=2,F\left(1\right)=3,F(x+2)=2F\left(x\right)-F(x+1)forx\ge 0$, then $F\left(5\right)$ is equal to

1. $-7$

2. $-3$

3. $17$

4. $13$

Q. The solution of the differential equation $\sqrt{a+x}\frac{dy}{dx}+xy=0$ (Where A is an arbitrary constant.)

1. $y=A{e}^{2/3(2a-x)\sqrt{x+a}}$
   
2. $y=A{e}^{-2/3(a-x)\sqrt{x+a}}$
   
3. $y=A{e}^{2/3(2a+x)\sqrt{x+a}}$
   
4. $y=A{e}^{-2/3(2a-x)\sqrt{x+a}}$

Q. The equation of the curve passing through (3, 9) which satisfies
​$\frac{dy}{dx}=\frac{x^3+1}{x^2}$is

1. $6xy=3x^3-6x+29$
2. $6xy=3x^2-29x+6$
3. $6xy=3x^3+29x-6$
4. $Noneofthese$

Q. The solution of $\frac{dy}{dx}=\frac{ax+b}{cy+d}$ represents a parabola if

1. $a=0,c=0$
2. $a=1,b=2$
3. $a=0,c\ne 0$
4. $a=1,c=1$

Q. The solution of the differential equation $x^4\frac{dy}{dx}+x^{3}y+cosec\left(xy\right)=0$ is equal to

1. $2cos\left(xy\right)+x^{-2}=c$
   
   
2. $2cos\left(xy\right)+y^{-2}=c$
   
3. $2sin\left(xy\right)+x^{-2}=c$
   
4. $2sin\left(xy\right)+y^{-2}=c$

Q. The curve passing through the point $(1,1)$ satisfies the differential equation $\frac{dy}{dx}+\frac{\sqrt{(x^2-1)(y^2-1)}}{xy}=0$. If the curve passes through the point $(\sqrt{2},k)$, then the largest value of $|\left[k\right]|$ is
(where, $[.]$ represents the greatest integer function)

Q. The differential equation $y\frac{dy}{dx}+x=a$ (a is any constant) represents

1. A set of circles having centre on the y-axis
2. A set of circles centre on the x-axis
3. A set of ellipses
4. None of these

Q. Solution to the differential equation $\frac{x+\frac{x^3}{3!}+\frac{x^5}{5!}+.....}{1+\frac{x^2}{2!}+\frac{x^4}{4!}+.....}=\frac{dx-dy}{dx+dy}$ is

1. $2y{e}^{2x}=C.e^{2x}+1$
   
2. $2y{e}^{2x}=C.e^{2x}-1$
   
3. $y{e}^{2x}=C.e^{2x}+2$
   
4. $2x{e}^{2y}=C.e^x-1$

Q. The solution of the differential equation $\frac{dy}{dx}-ky=0,y\left(0\right)=1$ approach zero when $x\to \infty$, if

1. K = 0
2. k > 0
3. k < 0
4. none of these

Q. Let f be a differentiable function such that $f^{\prime }\left(x\right)=f\left(x\right)+{\int }_0^{2}f\left(x\right)dx,f\left(0\right)=\frac{4-e^2}{3}$, then f(x) is

1. $e^x-\left(\frac{e^2-1}{3}\right)$
2. $e^x-\left(\frac{e^2-2}{3}\right)$
3. $e^x+\frac{e-[arg|z-1|]}{3}$
4. $2^{ex}-\frac{e^2-1}{\left[arg\right(-|z|\left)\right]}$ (where [.] greatest integer & z is a complex number)

Q. Let $y=y\left(x\right)$ be a solution of the differential equation, $\sqrt{1-x^2}\frac{dy}{dx}+\sqrt{1-y^2}=0,|x|<1$.
If $y\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}$, then $y\left(\frac{-1}{\sqrt{2}}\right)$ is equal to:

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

Q. Given that the slope of the tangent to a curve $y=y\left(x\right)$ at any point $(x,y)$ is $\frac{2y}{x^2}$. If the curve passes through the centre of the circle $x^2+y^2-2x-2y=0$, then its equation is :

1. $x{\log }_e|y|=2(x-1)$
2. $x{\log }_e|y|=(x-1)$
3. $x{\log }_e|y|=-2(x-1)$
4. $x^2{\log }_e|y|=-2(x-1)$

Q.

The solution of the equation
$\frac{dy}{dx}=\frac{3x-4y-2}{3x-4y-3}$ is

1. $(x-y{)}^2+C=log(3x-4y+1)$
2. $x-y+C=log(3x-4y+4)$
3. $x-y+C=log(3x-4y-3)$
4. $x-y+C=log(3x-4y+1)$

Q. The number of value(s) of $x$ satisfying the equation ${\tan }^{-1}2x+{\tan }^{-1}3x=\frac{\pi }{4}$ is

1. infinitely many
2. $0$
3. $1$
4. $2$

Q. Solution to the differential equation $\frac{x+\frac{x^3}{3!}+\frac{x^5}{5!}+.....}{1+\frac{x^2}{2!}+\frac{x^4}{4!}+.....}=\frac{dx-dy}{dx+dy}$ is

1. $2y{e}^{2x}=C.e^{2x}+1$
   
2. $2y{e}^{2x}=C.e^{2x}-1$
   
3. $y{e}^{2x}=C.e^{2x}+2$
   
4. $2x{e}^{2y}=C.e^x-1$

Q. The equation of the curve passing through the origin and satisfying the differential equation ${\left(\frac{dy}{dx}\right)}^2=(x-y{)}^2$, is

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

Q. If $y=y\left(x\right)$ is the solution of the differential equation, $e^y(\frac{dy}{dx}-1)=e^x$ such that $y\left(0\right)=0$, then $y\left(1\right)$ is equal to

1. ${\log }_{e}2$
2. $2e$
3. $2+{\log }_{e}2$
4. $1+{\log }_{e}2$

Q. Let $y=y\left(x\right)$ be a solution of the differential equation, $\sqrt{1-x^2}\frac{dy}{dx}+\sqrt{1-y^2}=0,|x|<1$.
If $y\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}$, then $y\left(\frac{-1}{\sqrt{2}}\right)$ is equal to:

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

Q. The general solution of $\frac{dy}{dx}=\frac{1-3y-3x}{1+x+y}$ is
(where ${}^{\prime }{c}^{\prime }$ is constant of integration)

1. $x+3y+2\ln |1+x+y|=c$
2. $x+y+2\ln |1-x-y|=c$
3. $3x+y+2\ln |1+x+y|=c$
4. $3x+y+2\ln |1-x-y|=c$

Q. The sum of squares of the perpendiculars drawn from the points (0, 1) and (0, -1) to any tangent to a curve is 2. Then, the equation of the curve is

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

Q.

The solution of the differential equation $\frac{dy}{dx}=1+x+y+xy$ is

1. $log(1+y)=x+\frac{x^2}{2}+c$

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

4. None of these

Q. The curve passing through the point $(1,1)$ satisfies the differential equation $\frac{dy}{dx}+\frac{\sqrt{(x^2-1)(y^2-1)}}{xy}=0$. If the curve passes through the point $(\sqrt{2},k)$, then the largest value of $|\left[k\right]|$ is
(where, $[.]$ represents the greatest integer function)

Q. The general solution of the differential equation $\frac{dy}{dx}+y{g}^{\prime }\left(x\right)=g\left(x\right).g^{\prime }\left(x\right)$, where $g\left(x\right)$ is a given function of x, is

1. $g\left(x\right)+log[1+y+g(x\left)\right]=c$
2. $g\left(x\right)+log[1+y-g(x\left)\right]=c$
3. $g\left(x\right)+log[1+y-g(x\left)\right]=c$
4. None of the above

Q. The solution of the equation $log\left(\frac{dy}{dx}\right)=ax+by$ is

1. $\frac{e^{by}}{b}=\frac{e^{ax}}{a}+c$
2. $\frac{e^{-by}}{-b}=\frac{e^{ax}}{a}+c$
   
3. $\frac{e^{-by}}{a}=\frac{e^{ax}}{a}+c$
4. None of these

Q. Can ${\sin }^{-1}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=x+y$ be solved using the variable separable method?(yes/no)
Ans :

Q. The solution of the differential equation, $\frac{dy}{dx}=(x-y{)}^2$, when $y\left(1\right)=1$, is :

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

Q. The solution of $\frac{x^{3}dx+y{x}^{2}dy}{\sqrt{x^2+y^2}}=ydx-xdy$, $y\left(1\right)=1$ is

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

Q. The solution of the equation $\frac{dy}{dx}=cos(x-y)$ is

1. $y+cot\left(\frac{x-y}{2}\right)=C$
2. $x+cot\left(\frac{x-y}{2}\right)=C$
3. $x+tan\left(\frac{x-y}{2}\right)=C$
4. None of these

Q. Can ${\sin }^{-1}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=x+y$ be solved using the variable separable method?(yes/no)
Ans :

Q. Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is known that the rate at which the water level drops is proportional to the square root of water depth y, where the constant of proportionality k (Where k >0) depends on the acceleration due to gravity and the geometry of the hole. if t is measured in minutes and $k=\frac{1}{15},$ then the time to drain the tank, if the water 4 meter deep to start with is -

1. 30
2. 45
3. 60
4. 80