Formation of a Differential Equation from a General Solution

Q. The degree of the differential equation satisfying the relation $\sqrt{1+x^2}+\sqrt{1+y^2}=\lambda (x\sqrt{1+y^2}-y\sqrt{1+x^2})$ is

1. 3
2. 2
3. none of these
4. 1

Q. If $ycosx+xcosy=\pi ,then{y}^{\prime \prime }\left(0\right)$ is

1. 1
2. $\pi$
3. 0
4. $-\pi$

Q.

The differential equation of the family of parabolas with focus at the origin and the x-axis as axis is

[EAMCET 2003]

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

2. $y{\left(\frac{dy}{dx}\right)}^2+4x\frac{dy}{dx}=4y$

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

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

Q.

The differential equation of family of parobalas with foci at the origin and axis along the X- axis, is

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

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

3. None of these

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

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. 0
3. 1
4. None of these

Q.

The differential equation of the family of curves $y=acos(x+b)$ is

[MP PET 1993]

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

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

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

4. None of these

Q. The differential equation whose general solution is given by,
$y=\left(c_{1}cos\right(x+c_2\left)\right)-\left(c_3{e}^{(-x+c_4)}\right)+\left(c_{5}sinx\right)$ where $c_1,c_2,c_3,c_4,c_5$ are arbitrary constants, is

1. $\frac{d^{4}y}{d{x}^4}-\frac{d^{2}y}{d{x}^2}+y=0$
2. $\frac{d^{3}y}{d{x}^3}+\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$
3. $\frac{d^{5}y}{d{x}^5}+y=0$
4. $\frac{d^{3}y}{d{x}^3}+\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}-y=0$

Q. The order of the differential equation of all tangent lines to the parabola $y=x^2$ is

1. 1
2. 2
3. 3
4. 4

Q.

The differential equation corresponding to primitive $y=e^{dx}$ is

or

The elimination of the arbitrary constant m from the equation $y=e^{mx}$ gives the differential equation

[MP PET 1995, 2000; Pb. CET 2000]

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

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

3. $\frac{dy}{dx}=\left(\frac{y}{x}\right)logy$

4. $\frac{dy}{dx}=\left(\frac{x}{y}\right)logx$

Q. The third derivative of a function f(x) vanishes for all x. If f(0)=1, f'1(1)=2 and f”(1) = – 1, then f(x) is equal to

1. $(-\frac{3}{2})x^2+3x+9$
2. $(-\frac{1}{2})x^2-3x+1$
3. $(-\frac{1}{2})x^2+3x+1$
4. $(-\frac{3}{2})x^2-7x+2$