Order of a Differential Equation

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

1. $(2-\sqrt{3})-{\log }_{e}2$
2. $(1-\sqrt{3})-{\log }_e(\sqrt{3}-1)$
3. $(1+\sqrt{3})-{\log }_e(\sqrt{3}-1)$
4. $(2+\sqrt{3})+{\log }_{e}2$

Q. Let $y=y\left(x\right)$ be the solution of the differential equation $(x-x^3)dy=(y+y{x}^2-3x^4)dx,$ $x>2.$ If $y\left(3\right)=3$, then $y\left(4\right)$ is equal to

1. $12$
2. $8$
3. $4$
4. $16$

Q.

The order and degree of the differential equation$y=x\left(\frac{dy}{dx}\right)+\frac{2}{\left(\frac{dy}{dx}\right)}$, are

1. $1,2$

2. $1,3$

3. $2,1$

4. $1,1$

Q. Let $y=y\left(x\right)$ be the solution of the differential equation $\frac{dy}{dx}=2(y+2\sin x-5)x-2\cos x$ such that $y\left(0\right)=7$. Then $y\left(\pi \right)$ is equal to

1. $e^{{\pi }^2}+5$
2. $2e^{{\pi }^2}+5$
3. $7e^{{\pi }^2}+5$
4. $3e^{{\pi }^2}+5$

Q.

The order of the differential equation whose solution is $x^2+y^2+2gx+2fy+c=0$

1. $1$

2. $2$

3. $3$

4. $4$

Q.

Define coaxial circles and deduce their equation in the simplest form.

Q.

Which of the following is a homogeneous differential equation?

A.

B.

C.

D.

Q. What is meant by "......The given differential equation is not polynomial equation in its derivatives and so it's degree is not defined"

Q.

How many months does calculus takes to cover in class 11 (approx) ?

Q.

The solution of the differential equation $(x^2+y^2)dx-2xydy=0$ is

1. $x^2-y^2=Cx$

2. $x^2-y^2=Cy$

3. $x^2+y^2=Cx$

4. $x^2+y^2=Cy$

Q.

Write an equation describing the relationship of the given variables.

$y$ varies directly as the fourth power of $x$ and when $x=3,y=162$.

Q. Solution of differential equation $(y+x\sqrt{xy}(x+y\left)\right)dx+\left(y\sqrt{xy}\right(x+y)-x)dy=0$ is

1. $\frac{x^2+y^2}{2}+{\cot }^{-1}\sqrt{\frac{x}{y}}=\lambda$
2. $\frac{x^2+y}{2}-2{\cot }^{-1}\sqrt{\frac{x}{y}}=\lambda$
3. $\frac{x^2+y^2}{2}+{\tan }^{-1}\sqrt{\frac{y}{x}}=\lambda$
4. $\frac{x^2+y^2}{2}+{\tan }^{-1}\sqrt{\frac{y}{x}}=\lambda$

Q.

Solution of the differential equation $\left(2x-10y^3\right)\frac{dy}{dx}+y=0$ is

Q.

If $y^2=P\left(x\right)$, a polynomial of degree $3$, then $2\frac{d}{dx}\left(y^3\frac{d^{2}y}{d{x}^2}\right)$ equals

1. $P\left(x\right)+P\left(x\right)$

2. $P\left(x\right)P\left(x\right)$

3. $P\left(x\right)P\left(x\right)$

4. a constant

Q. Order of a differential equation is the highest power of any derivative appearing in the differential equation

1. True
2. False

Q.

Solve the differential equation
$y{e}^{\frac{x}{y}}dx=(x{e}^{\frac{x}{y}}+y^2)dy(y\ne 0)$

Q.

The order of the differential equation of the curve y=ax^2+bx+c where a, b and 'c' are arbitrary constants is

Q.

The degree of the differential equation satisfying $\sqrt{1-x^2}+\sqrt{1-y^2}=a\left(x-y\right)$ is

1. $1$

2. $3$

3. $2$

4. None of these

Q.

The order and degree of the differential equation $\left(\sqrt{\frac{dy}{dx}}\right)-4\left(\frac{dy}{dx}\right)-7x=0$ are

1. $1$ and $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

2. $2$ and $1$

3. $1$ and $1$

4. $1$ and $2$

Q.

The order of a differential equation representing a family of curves is same as:

1. The degree of the equation

2. The number of arbitrary constants present in the equation.

3. The number of curves in the family

4. The number of variables in the equation

Q. Statement 1: The differential equation $y^{3}dy+(x+y^2)dx=0$ becomes homogeneous if we put $y^2=t$
Statement 2: All differential equation of first order and first degree of curves $f(x,y)=0$ becomes homogeneous if we put $y=tx$

1. Statement -1 is True, Statement-2 is True; Statement-2 is a correct explaination for Statement-1.
2. Statement -1 is True, Statement-2 is True; Statement-2 is NOT a correct explaination for Statement-1.
3. Statement -1 is True, Statement-2 is False
4. Statement -1 is False, Statement-2 is True.

Q. The solution of the differential equation $\cos xdy=y\left(\sin \right(x)-y)dx,$ $0<x<\frac{\pi }{2}$ is
$($Here, $C$ is a constant of integration$)$

1. $y\sec x=\tan x+C$
2. $y\tan x=\sec x+C$
3. $\tan x=(\sec x+C)y$
4. $\sec x=(\tan x+C)y$

Q. The order and degree of the differential equation ${[1+{\left(\frac{dy}{dx}\right)}^2+\sin \left(\frac{dy}{dx}\right)]}^{3/4}=\frac{d^{2}y}{d{x}^2}$ are:

1. order$=2$; degree $=3$
2. order$=2$; degree $=4$
3. order$=2$; degree $=\frac{3}{4}$
4. order$=2$; degree $=$ not defined

Q. The degree of the differential equation ${\left(\frac{d^{2}y}{d{x}^2}\right)}^3+{\left(\frac{dy}{dx}\right)}^2+\sin {\left(\frac{dy}{dx}\right)}^2+\sin \left(\frac{dy}{dx}\right)+1=0$ is

1. $3$
2. $2$
3. $1$
4. None of the above

Q. Consider the differential equation $\frac{d^{2}y}{d{x}^2}={(1+\frac{dy}{dx})}^{2/3}$. Find the degree and order of the equation.

Q. Find the order and degree of the differential equation $\frac{d^{2}y}{d{x}^2}={[1+{\left(\frac{dy}{dx}\right)}^2]}^{\frac{3}{2}}$.

Q. Find the order and degree, if defined of the differential equation
${\left(\begin{array}{c}\frac{d^{2}y}{d{x}^2}\end{array}\right)}^3+{\left(\begin{array}{c}\frac{dy}{dx}\end{array}\right)}^2+\sin \frac{dy}{dx}+1=0$.

Q. Consider the following statements in respect of the differential equation $\frac{d^{2}y}{d{x}^2}+cos\left(\frac{dy}{dx}\right)=0:$
1. The degree of the differential equation is not defined.
2. The order of the differential equation is 2.
Which of the above statements is/are correct ?

1. 1 only
2. 2 only
3. Both 1 and 2
4. not defined

Q. The integrating factor of the differential equation $\cos x\frac{dy}{dx}+\sin x\cdot y=x\cdot \sec x,(\cos x\ne 0)$ is

1. $\sec x$
2. $\tan x$
3. $\cos x$
4. $\sin x$

Q. The order and degree of the differential equation ${[1+{\left(\frac{dy}{dx}\right)}^2]}^3={\rho }^2{\left[\frac{d^{2}y}{d{x}^2}\right]}^2$ are respectively.

1. $3$ and $2$
2. $2$ and $2$
3. $2$ and $3$
4. $1$ and $3$