Differential Equations · NDA previous year questions

Passage: Consider the differential equation $e^{x+y}\frac{dy}{dx}=e^{x-y}$. Question: What is the solution of the differential equation with $y(0)=0$ ?

1. A.$y=\ln (2x+1)$
2. B.$y=\ln (2x-1)$
3. C.$2y=\ln (2x+1)$
4. D.$2y=\ln (2x-1)$

Passage: Consider the differential equation $e^{x+y}\frac{dy}{dx}=e^{x-y}$. Question: What is $\frac{d^{2}y}{d{x}^2}{\left(\frac{dx}{dy}\right)}^2$ equal to ?

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

1. A.$\cos \left(\frac{y-q}{x}\right)=p$
2. B.$\cos \left(\frac{y-p}{x}\right)=q$
3. C.${\cos }^{-1}\left(\frac{y-q}{x}\right)=p$
4. D.${\cos }^{-1}\left(\frac{y-p}{x}\right)=q$

What is the degree of the differential equation

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

Direction : Consider the following for the items that follow : Let $ydx+(x-y^3)dy=0$ be a differential equation. What are the order and degree respectively of the differential equation?

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

Direction : Consider the following for the items that follow : Let $ydx+(x-y^3)dy=0$ be a differential equation. What is the solution of the differential equation?

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

Direction : Consider the following for the items that follow : Let $f(x)=|x^2-x-2|$ What is ${\int }_0^{2}f(x)dx$ equal to

1. A.0
2. B.1
3. C.5/3
4. D.10/3

Direction : Consider the following for the items that follow : Let $f(x)=|x^2-x-2|$ What is ${\int }_1^{3}f(x)dx$ equal to

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

Let $y_1(x)$ and $y_2(x)$ be two solutions of the differential equation $\frac{dy}{dx}=x$. If $y_1(0)=0$ and $y_2(0)=4$, then what is the number of points of intersection of the curves $y_1(x)$ and $y_2(x)$?

1. A.No point
2. B.One point
3. C.Two points
4. D.More than two points

The differential equation, representing the curve $y=e^x(a\cos x+b\sin x)$ where $a$ and $b$ are arbitrary constants, is

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

If $f(x)=ax-b$ and $g(x)=cx+d$ are such that $f(g(x))=g(f(x))$, then which one of the following holds?

1. A.$f(d)=g(b)$
2. B.$f(b)+g(d)=0$
3. C.$f(a)+g(c)=2a$
4. D.$f(d)+g(b)=2d$

What is ${\int }_{-1}^1(3\sin x-\sin 3x){\cos }^{2}xdx$ equal to?

1. A.$\frac{1}{4}$
2. B.0
3. C.$\frac{1}{2}$
4. D.$-\frac{1}{4}$

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

If $\frac{dy}{dx}=2e^x{y}^3$, $y(0)=\frac{1}{2}$ then what is $4y^2(2-e^x)$ equal to?

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

Let $p={\int }_a^{b}f(x)dx$ and $q={\int }_a^b|f(x)|dx$. If $f(x)=e^{-x}$, then which one of the following is correct ?

1. A.$p=2q$
2. B.$p=-q$
3. C.$4p=q$
4. D.$p=q$

What is ${\int }_0^{\pi /2}\frac{a+\sin x}{2a+\sin x+\cos x}dx$ equal to?

1. A.$\frac{\pi }{4}$
2. B.$\frac{\pi }{2}$
3. C.1
4. D.0

Consider the following for the next items that follow: A parabola passes through (1, 2) and satisfies the differential equation $\frac{dy}{dx}=\frac{2y}{x},x>0,y>0$. What is the directrix of the parabola ?

1. A.$y=-\frac{1}{8}$
2. B.$y=\frac{1}{8}$
3. C.$x=-\frac{1}{8}$
4. D.$x=\frac{1}{8}$

What are the order and the degree respectively of the differential equation $x^2{\left(\frac{d^{3}y}{d{x}^3}\right)}^2+{\left(\frac{dy}{dx}\right)}^4+\sin x=0$ ?

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

What is the differential equation of all parabolas of the type $y^2=4a(x-b)$?

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

Consider the following for the next two (02) items that follow : Suppose E is the differential equation representing family of curves $y^2=2cx+2c\sqrt{c}$ where c is a positive parameter. What is the order of the differential equation ?

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

1. A.2
2. B.3
3. C.4
4. D.Degree does not exist

Consider the following for the next three (03) items that follow : Let $f(x)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|$ What is f(0) equal to ?

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

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

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

A solution of the differential equation ${\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2-x\frac{\mathrm{dy}}{\mathrm{dx}}=0$ is

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