Formation of a Differential Equation from a General Solution

Q.

Form the differential equation of the family of hyperbolas having foci on the x-axis and center at the origin.

Q.

The differential equation of all the lines in the xy-plane is

1. $\frac{dy}{dx}-x=0$

2. $\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}=0$

3. $\frac{d^{2}y}{d{x}^2}=0$

4. $\frac{d^{2}y}{d{x}^2}+x=0$

Q.

The normal to the curve $y(x-2)(x-3)=x+6$ at the point where the curve intersects the $y$-axis passes through the point

1. $\left(\frac{1}{2},\frac{1}{2}\right)$

2. $\left(\frac{1}{2},\frac{-1}{3}\right)$

3. $\left(\frac{1}{2},\frac{1}{3}\right)$

4. $\left(\frac{-1}{2},\frac{-1}{2}\right)$

Q.

The differential equation of the family of curves $y^2=4a(x+a)$, where a is an arbitrary constant, is

1. $y[1+{\left(\frac{dy}{dx}\right)}^2]=2x\frac{dy}{dx}$
   

2. $y[1-{\left(\frac{dy}{dx}\right)}^2]=2x\frac{dy}{dx}$
   

3. $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}=0$

4. ${\left(\frac{dy}{dx}\right)}^3+3\frac{dy}{dx}+y=0$

Q. An equation of the curve in which subnormal varies as the square of the ordinate is (k is constant of proportionality)

1. $y=A{e}^{kx}$
2. $y=e^{kx}$
3. $\frac{y^2}{2}+kx=A$
4. $y^2+k{x}^2=A$

Q. The equation of the curve which passes through the point (1, 1) and whose slope is given by $\frac{2y}{x}$, is

1. $y=x^2$
2. $x^2-y^2=0$
3. $2x^2+y^2=3$
4. $Noneofthese$

Q. The differential equation of all circles which pass through the origin and whose centers lie on the y-axis is

1. $(x^2-y^2)\frac{dy}{dx}-2xy=0$
2. $(x^2-y^2)\frac{dy}{dx}+2xy=0$
3. $(x^2-y^2)\frac{dy}{dx}-xy=0$
4. $(x^2-y^2)\frac{dy}{dx}+xy=0$

Q. The normal to a curve at P(x, y) meets the x-axis at G. If the distance of G from the origin is twice the abscissa of P, then the curve is a

1. ellipse
2. parabola
3. circle
4. hyperbola

Q.

If a real value of function $f$of a real variable $x$is such that $\frac{1}{(1+x)(1+x^2)}=\frac{A}{(1+x)}+\frac{f\left(x\right)}{(1+x^2)}$then $f\left(x\right)=$

1. $\frac{(1-x)}{2}$

2. $\frac{(x^2+1)}{2}$

3. $1-x$

4. None of these

Q. The differential equation satisfied by the curves $e^{2y}+2bx{e}^y+b^2=0,$ where $b$ is a parameter, is

1. $(1+x^2)\frac{d^{2}y}{d{x}^2}=0$
2. $(x^2-1){\left(\frac{dy}{dx}\right)}^2-1=0$
3. $(x^2-1)\frac{d^{2}y}{d{x}^2}+2=0$
4. $(x^2-x)\frac{dy}{dx}-y^2=0$

Q. If $x^2+y^2=t-\frac{1}{t}$ and $x^4+y^4=t_2+\frac{1}{t^2}$ , then $x^{3}y\frac{dy}{dx}$ equals

1. – 1
2. \N
3. 1
4. None of these

Q. The sum of squares of the length of the perpendiculars drawn from the points $(0,1)$ and $(0,-1)$ to any tangent to a curve $y=f\left(x\right)$ is $2$ units. Then the number of roots of $f\left(x\right)=0$ is

Q. The differential equation formed by eliminating parameters A and B from the family of curves given by$y=A{e}^{2x}+B{e}^{-2x}$ will be of ___ order.

1. First
2. Second
3. Third
4. None of the above

Q. If $y^2=a{x}^2+bx+c$, then $y^3.\frac{d^{2}y}{d{x}^2}$ is

1. A constant
2. A function of x only
3. A function of y only
4. A function of x and y

Q.

The differential equation of the family of curves $y^2=4a(x+a)$, where a is an arbitrary constant, is

1. $y[1+{\left(\frac{dy}{dx}\right)}^2]=2x\frac{dy}{dx}$
   

2. $y[1-{\left(\frac{dy}{dx}\right)}^2]=2x\frac{dy}{dx}$
   

3. $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}=0$

4. ${\left(\frac{dy}{dx}\right)}^3+3\frac{dy}{dx}+y=0$

Q.

Tangent to a curve intercepts the y-axis at a point $P$. A line perpendicular to this tangent through $P$ passes through another point (1, 0) the differential equation of the curve is

1. $y\frac{dy}{dx}-x{\left(\frac{dy}{dx}\right)}^2=1$

2. $y\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2=0$

3. $y\frac{dx}{dy}+x=1$

4. None of these

Q.

$y=4sin3x$ is a solution of the differential equation

[AI CBSE 1986]

1. $\frac{dy}{dx}+8y=0$

2. $\frac{dy}{dx}-8y=0$

3. $\frac{d^{2}y}{d{x}^2}+9y=0$

4. $\frac{d^{2}y}{d{x}^2}-9y=0$

Q. If tangents are drawn from a point $P(2,0)$ to the curve $\sqrt{1+y^2}=\frac{x}{3}$ which meet the curve at $A$ and $B,$ then

1. acute angle between the tangents is ${\tan }^{-1}\left(\frac{\sqrt{5}}{2}\right)$
2. area of $△PAB$ is $\frac{5\sqrt{5}}{4}$ sq. units
3. $△PAB$ is equilateral
4. $AB=\sqrt{5}$

Q. The DE for the family of circles having a given radius 'a' is same as the DE of the family of circles whose center lies at origin.

1. False
2. True

Q. The order of the differential equation of all circles of a given radius a is.

Q. If $x^2+y^2=t-\frac{1}{t}$ and $x^4+y^4=t_2+\frac{1}{t^2}$ , then $x^{3}y\frac{dy}{dx}$ equals

1. – 1
2. \N
3. 1
4. None of these