Differential Equations · NDA previous year questions

What is the solution of the differential equation $\ln \left(\frac{dy}{dx}\right)-a=0$?

1. A.$y=x{e}^a+c$
2. B.$x=y{e}^a+c$
3. C.$y=\ln x+c$
4. D.$x=\ln y+c$

What are the degree and order respectively of the differential equation satisfying $e^{y\sqrt{1-x^2}+x\sqrt{1-y^2}}=c{e}^x$, (where $c>0$, $|x|<1$, $|y|<1$)?

1. A.1, 1
2. B.1, 2
3. C.2, 1
4. D.2, 2

What is the order of the differential equation $\frac{dx}{dy}+\int ydx=x^3$?

1. A.1
2. B.2
3. C.3
4. D.Cannot be determined

Which one of the following differential equations represents the family of straight lines which are at unit distance from the origin?

1. A.${\left(y-x\frac{dy}{dx}\right)}^2=1-{\left(\frac{dy}{dx}\right)}^2$
2. B.${\left(y+x\frac{dy}{dx}\right)}^2=1+{\left(\frac{dy}{dx}\right)}^2$
3. C.${\left(y-x\frac{dy}{dx}\right)}^2=1+{\left(\frac{dy}{dx}\right)}^2$
4. D.${\left(y+x\frac{dy}{dx}\right)}^2=1-{\left(\frac{dy}{dx}\right)}^2$

Consider the following for the next two (02) items that follow : Let $\vec{a}=\hat{i}+\hat{j}$, $\vec{b}=3\hat{i}+4\hat{k}$ and $\vec{b}=\vec{c}+\vec{d}$, where $\vec{c}$ is parallel to $\vec{a}$ and $\vec{d}$ is perpendicular to $\vec{a}$. What is $\vec{c}$ equal to?

1. A.$\frac{3(\hat{i}+\hat{j})}{2}$
2. B.$\frac{2(\hat{i}+\hat{j})}{3}$
3. C.$\frac{\hat{i}+\hat{j}}{2}$
4. D.$\frac{\hat{i}+\hat{j}}{3}$

Consider the following for the next two (02) items that follow : Let $\vec{a}=\hat{i}+\hat{j}$, $\vec{b}=3\hat{i}+4\hat{k}$ and $\vec{b}=\vec{c}+\vec{d}$, where $\vec{c}$ is parallel to $\vec{a}$ and $\vec{d}$ is perpendicular to $\vec{a}$. If $\vec{d}=x\hat{i}+y\hat{j}+z\hat{k}$, then which of the following equations is/are correct? 1. $y-x=4$ 2. $2z-3=0$ Select the correct answer using the code given below :

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

What are the order and degree respectively of the differential equation whose solution is $y=cx+c^2-3c^{3/2}+2$, where $c$ is a parameter?

1. A.1, 2
2. B.2, 2
3. C.1, 3
4. D.1, 4

The degree of the differential equation $$\frac{dy}{dx}-x={\left(y-x\frac{dy}{dx}\right)}^{-4}\text{ is}$$

1. A.2
2. B.3
3. C.4
4. D.5

The solution of $\frac{dy}{dx}=\sqrt{1-x^2-y^2+x^2{y}^2}$ is where $c$ is an arbitrary constant

1. A.${\sin }^{-1}y={\sin }^{-1}x+c$
2. B.$2{\sin }^{-1}y=\sqrt{1-x^2}+{\sin }^{-1}x+c$
3. C.$2{\sin }^{-1}y=x\sqrt{1-x^2}+{\sin }^{-1}x+c$
4. D.$2{\sin }^{-1}y=x\sqrt{1-x^2}+{\cos }^{-1}x+c$

The differential equation of the family of circles passing through the origin and having centres on the $x$-axis is

1. A.$2xy\frac{dy}{dx}=x^2-y^2$
2. B.$2xy\frac{dy}{dx}=y^2-x^2$
3. C.$2xy\frac{dy}{dx}=x^2+y^2$
4. D.$2xy\frac{dy}{dx}+x^2+y^2=0$

The order and degree of the differential equation of parabolas having vertex at the origin and focus at $(a,0)$ where $a>0$, are respectively

1. A.1, 1
2. B.2, 1
3. C.1, 2
4. D.2, 2

The area of the square, one of whose diagonals is $3\hat{i}+4\hat{j}$ is

1. A.12 square unit
2. B.12.5 square unit
3. C.25 square unit
4. D.156.25 square unit

What is the solution of the differential equation $\frac{ydx-xdy}{y^2}=0$? where $c$ is an arbitrary constant.

1. A.$xy=c$
2. B.$y=cx$
3. C.$x+y=c$
4. D.$x-y=c$

What is the solution of the differential equation $\sin \left(\frac{dy}{dx}\right)-a=0$? where $c$ is an arbitrary constant.

1. A.$y=x{\sin }^{-1}a+c$
2. B.$x=y{\sin }^{-1}a+c$
3. C.$y=x+x{\sin }^{-1}a+c$
4. D.$y={\sin }^{-1}a+c$

What is the solution of the differential equation $\frac{dx}{dy}+\frac{x}{y}-y^2=0$? where $c$ is an arbitrary constant.

1. A.$xy=x^4+c$
2. B.$xy=y^4+c$
3. C.$4xy=y^4+c$
4. D.$3xy=y^3+c$

What is the degree of the differential equation ${\left(\frac{d^{3}y}{d{x}^3}\right)}^{3/2}={\left(\frac{d^{2}y}{d{x}^2}\right)}^2$?

1. A.1
2. B.2
3. C.3
4. D.4

What is the solution of the equation $\ln \left(\frac{dy}{dx}\right)+x=0$? where c is an arbitrary constant.

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

Eliminating the arbitrary constants B and C in the expression $y=\frac{2}{3C}(Cx-1{)}^{3/2}+B$, we get

1. A.$x\left[1+{\left(\frac{dy}{dx}\right)}^2\right]=\frac{d^{2}y}{d{x}^2}$
2. B.$2x\left(\frac{dy}{dx}\right)\frac{d^{2}y}{d{x}^2}=1+{\left(\frac{dy}{dx}\right)}^2$
3. C.$\left(\frac{dy}{dx}\right)\frac{d^{2}y}{d{x}^2}=1$
4. D.${\left(\frac{dy}{dx}\right)}^2+1=\frac{d^{2}y}{d{x}^2}$

If $|\vec{a}|=2$, $|\vec{b}|=5$ and $|\vec{a}\times \vec{b}|=8$, then what is $\vec{a}\cdot \vec{b}$ equal to?

1. A.6
2. B.7
3. C.8
4. D.9

What is the number of arbitrary constants in the particular solution of differential equation of third order?

1. A.0
2. B.1
3. C.2
4. D.3

What is the general solution of the differential equation $xdy-ydx=y^2$? where c is an arbitrary constant

1. A.$x=cy$
2. B.$y^2=cx$
3. C.$x+xy-cy=0$
4. D.None of the above

The solution of $\frac{dy}{dx}=|x|$ is:

1. A.$y=\frac{x|x|}{2}+c$
2. B.$y=\frac{|x|}{2}+c$
3. C.$y=\frac{x^2}{2}+c$
4. D.$y=\frac{x^3}{2}+c$

What is the solution of $\frac{dy}{dx}+2y=1$ satisfying $y(0)=0$?

1. A.$y=\frac{1-e^{-2x}}{2}$
2. B.$y=\frac{1+e^{-2x}}{2}$
3. C.$y=1+e^x$
4. D.$y=\frac{1+e^x}{2}$

What is the equation of a curve passing through $(0,1)$ and whose differential equation is given by $dy=y\tan xdx$?

1. A.$y=\cos x$
2. B.$y=\sin x$
3. C.$y=\sec x$
4. D.$y=\mathrm{cosec}x$

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. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2