Differential Equations · NDA previous year questions

If $f(x)=x^2+2$ and $g(x)=2x-3$, then what is $(f\circ g)(1)$ equal to ?

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

Consider the following statements: 1. The degree of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}+\cos \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=0$ is 1. 2. The order of the differential equation ${\left(\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}\right)}^3+\cos \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=0$ is 2. Which of the statements above is/are correct?

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

What is the differential equation of the family of parabolas having a vertex at origin and axis along positive y-axis?

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

What is the solution of the differential equation (dy − dx) + cos x(dy + dx) = 0 ?

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

What is the degree of the differential equation $1+{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}^4={\left(\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}\right)}^{\frac{4}{3}}$?

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

A radioactive substance decays at a rate proportional to the amount of substance present. If half of the substance decays in 100 years, then what is the decay constant (proportionality constant) ?

1. A.$\frac{\ln 2}{100}$
2. B.$\frac{\ln 5}{100}$
3. C.$\frac{\ln 10}{100}$
4. D.$\frac{2\ln 2}{100}$

What is the order of the differential equation of all ellipses whose axes are along the coordinate axes?

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

What is the degree of the differential equation of all circles touching both the coordinate axes in the first quadrant?

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

What is the differential equation of $y=A-\frac{B}{x}$?

1. A.$x{y}_2+y_1=0$
2. B.$x{y}_2+2y_1=0$
3. C.$x{y}_2-2y_1=0$
4. D.$2x{y}_2+y_1=0$

What is ${\int }_0^{\pi }\ln \left(\tan \frac{x}{2}\right)\mathrm{d}x$ equal to?

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

If the general solution of a differential equation is $y^2+2cy-cx+c^2=0$, where $c$ is an arbitrary constant, then what is the order of the differential equation?

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

What is the degree of the following differential equation?

1. A.1
2. B.2
3. C.3
4. D.Degree is not defined

Which one of the following differential equations has the general solution $y=a{e}^x+b{e}^{-x}$?

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

What is the solution of the following differential equation? $\ln \left(\frac{dy}{dx}\right)+y=x$

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

What is $\int e^{(2\ln x+\ln x^2)}\mathrm{d}x$ equal to?

1. A.$\frac{x^4}{4}+c$
2. B.$\frac{x^3}{3}+c$
3. C.$\frac{2x^5}{5}+c$
4. D.$\frac{x^5}{5}+c$

The order and degree of the differential equation $k\frac{dy}{dx}=\int {\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^{\frac{2}{3}}dx$ are respectively

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

What is ${\lim }_{x\to 0}\frac{\sin x\log (1-x)}{x^2}$ equal to?

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

If $f(x)=3x^2-5x+p$ and $f(0)$ and $f(1)$ are opposite in sign, then which of the following is correct?

1. A.$-2<p<0$
2. B.$-2<p<2$
3. C.$0<p<2$
4. D.$3<p<5$

The solution of the differential equation $dy=(1+y^2)dx$ is

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

What is $\int (e^{\log x}+\sin x)\cos xdx$ equal to?

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

The function $u(x,y)=c$ which satisfies the differential equation $x(dx-dy)+y(dy-dx)=0$, is

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

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

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

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

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

Which one of the following is the differential equation that represents the family of curves $y=\frac{1}{2x^2-c}$ where $c$ is an arbitrary constant?

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

The differential equation which represents the family of curves given by $\tan y=c(1-e^x)$ is

1. A.$e^x\tan ydx+(1-e^x)dy=0$
2. B.$e^x\tan ydx+(1-e^x){\sec }^{2}ydy=0$
3. C.$e^x(1-e^x)dx+\tan ydy=0$
4. D.$e^x\tan ydy+(1-e^x)dx=0$