Differential Equations · NDA previous year questions

For the next three (03) items that follow: The general solution of the differential equation $(x^2+x+1)dy+(y^2+y+1)dx=0$ is $(x+y+1)=A(1+Bx+Cy+Dxy)$ where B, C and D are constants and A is parameter. What is B equal to?

1. A.$-1$
2. B.1
3. C.2
4. D.None of the above

For the next three (03) items that follow: The general solution of the differential equation $(x^2+x+1)dy+(y^2+y+1)dx=0$ is $(x+y+1)=A(1+Bx+Cy+Dxy)$ where B, C and D are constants and A is parameter. What is C equal to?

1. A.1
2. B.$-1$
3. C.2
4. D.None of the above

For the next three (03) items that follow: The general solution of the differential equation $(x^2+x+1)dy+(y^2+y+1)dx=0$ is $(x+y+1)=A(1+Bx+Cy+Dxy)$ where B, C and D are constants and A is parameter. What is D equal to?

1. A.$-1$
2. B.1
3. C.$-2$
4. D.None of the above

What is the order of the differential equation ${\left(\frac{dy}{dx}\right)}^2+\frac{dy}{dx}-{\sin }^{2}y=0$ ?

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

$y=2\cos x+3\sin x$ satisfies which of the following differential equations? 1. $\frac{d^{2}y}{d{x}^2}+y=0$ 2. ${\left(\frac{dy}{dx}\right)}^2+\frac{dy}{dx}=0$ 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

The differential equation of all circles whose centres are at the origin is

1. A.$\frac{dy}{dx}=\frac{y}{x}$
2. B.$\frac{dy}{dx}=\frac{x}{y}$
3. C.$\frac{dy}{dx}=-\frac{x}{y}$
4. D.None of the above

What is $\int {\sin }^{2}xdx+\int {\cos }^{2}xdx$ equal to? where $c$ is an arbitrary constant.

1. A.$x+c$
2. B.$\frac{x^2}{2}+c$
3. C.$x^2+c$
4. D.None of the above

What is $\int e^{e^x}{e}^{x}dx$ equal to? where $c$ is an arbitrary constant.

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

The differential equation of the curve $y=\sin x$ is

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

The degree and order respectively of the differential equation $\frac{dy}{dx}=\frac{1}{x+y+1}$ are

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

What is the degree of the differential equation ${\left(\frac{d^{4}y}{d{x}^4}\right)}^{3/5}-5\frac{d^{3}y}{d{x}^3}+6\frac{d^{2}y}{d{x}^2}-8\frac{dy}{dx}+5=0$ ?

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

The general solution of the differential equation $x\frac{dy}{dx}+y=0$ is where $c$ is an arbitrary constant.

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

What is the area of the triangle bounded by the sides $x=0,y=0$ and $x+y=2$ ?

1. A.1 square unit
2. B.2 square unit
3. C.4 square unit
4. D.8 square unit

The general solution of the differential equation $\ln \left(\frac{dy}{dx}\right)+x=0$ is 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.$y=-e^x+c$

If $\vec{\beta }$ is perpendicular to both $\vec{\alpha }$ and $\vec{\gamma }$ where $\vec{\alpha }=\hat{k}$ and $\vec{\gamma }=2\hat{i}+3\hat{j}+4\hat{k}$, then what is $\vec{\beta }$ equal to ?

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

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

1. A.6
2. B.3
3. C.2
4. D.1

Consider a differential equation of order m and degree n. Which one of the following pairs is not feasible?

1. A.(3, 2)
2. B.(2, 3/2)
3. C.(2, 4)
4. D.(2, 2)

The differential equation representing the family of curves $y=a\sin (\lambda x+\alpha )$ is:

1. A.$\frac{d^{2}y}{d{x}^2}+{\lambda }^{2}y=0$
2. B.$\frac{d^{2}y}{d{x}^2}-{\lambda }^{2}y=0$
3. C.$\frac{d^{2}y}{d{x}^2}+\lambda y=0$
4. D.None of the above

The differential equation $y\frac{dy}{dx}+x=a$ where ‘a’ is any constant represents:

1. A.A set of straight lines
2. B.A set of ellipses
3. C.A set of circles
4. D.None of the above

For the differential equation ${\left(\frac{dy}{dx}\right)}^2-x\left(\frac{dy}{dx}\right)+y=0$, which one of the following is not its solution?

1. A.$y=x-1$
2. B.$4y=x^2$
3. C.$y=x$
4. D.$y=-x-1$

What is the general solution of the differential equation $x^{2}dy+y^{2}dx=0$? where c is the constant of integration

1. A.$x+y=c$
2. B.$xy=c$
3. C.$c(x+y)=xy$
4. D.None of the above

What is the general solution of the differential equation $e^x\tan ydx+(1-e^x){\sec }^{2}ydy=0$? where c is the constant of integration

1. A.$\sin y=c(1-e^x)$
2. B.$\cos y=c(1-e^x)$
3. C.$\cot y=c(1-e^x)$
4. D.None of the above

Which one of the following differential equations is not linear?

1. A.$\frac{d^{2}y}{d{x}^2}+4y=0$
2. B.$x\frac{dy}{dx}+y=x^3$
3. C.$(x-y{)}^2\frac{dy}{dx}=9$
4. D.${\cos }^{2}x\frac{dy}{dx}+y=\tan x$

What is the solution of the differential equation $\frac{dy}{dx}+\frac{y}{x}=0$ ? where c is a constant

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

What is the degree of the differential equation $y=x\frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^{-1}$ ?

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