Order and Degree

Q.

If $\mathrm{y}={\left(\mathrm{x}+\sqrt{1+{\mathrm{x}}^2}\right)}^{\mathrm{n}}$, then $\left(1+{\mathrm{x}}^2\right)\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}+\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}$ is

1. ${\mathrm{n}}^2\mathrm{y}$

2. $-{\mathrm{n}}^2\mathrm{y}$

3. $-\mathrm{y}$

4. $2{\mathrm{x}}^2\mathrm{y}$

Q.

The order and degree of differential equation ${\left(\frac{d^{3}y}{d{x}^3}\right)}^{\frac{4}{5}}-2\left(\frac{dy}{dx}\right){\left(\frac{d^{2}y}{d{x}^2}\right)}^2=0$ are

1. $2,10$

2. $3,10$

3. $3,5$

4. $3,4$

Q. The partial differential equation $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\frac{{\partial }^{2}u}{\partial x^2}$ is a

1. Linear equation of order $2$
2. Non-linear equation of order $1$
3. Linear equation of order $1$
4. Non-linear equation of order $2$

Q. Choose the CORRECT set of functions, which are linearly dependent.

1. sin x, $si{n}^{2}xandco{s}^{2}x$
2. $cosx,sinx,andtanx$
3. cos 2x, $si{n}^{2}xandco{s}^{2}x$
4. cos 2x, sin x and cos x

Q. The differential $\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+siny=0$ is

1. linear
2. non-linear
3. homogeneous
4. of degree two

Q. Consider the following statements about the linear dependence of the real valued functions $y_1=1,y_2=x$ and $y_3=x^2$, over the field of real numbers.

I. $y_1,y_2$ and $y_3$ are linearly independent on $-1\le x\le 0$

II. $y_1,y_2$ and $y_3$ are linearly dependent on $0\le x\le 1$

III. $y_1,y_2$ and $y_3$ are linearly dependent on $0\le x\le 1$

IV. $y_1,y_2$ and $y_3$ are linearly independent on $-1\le x\le 1$

Which on among the following is correct?

1. Both I and IV are True
2. Both I and III are True
3. Both II and IV are True
4. Both III and IV are True

Q. The Blasius equation, $\frac{d^3}{d{\eta }^3}+\frac{f{d}^{2}f}{2d{\eta }^2}=0$, is a

1. second order nonlinear ordinary differential equation
2. third order nonlinear ordinary differential equation
3. third order linear ordinary differential equation
4. mixed order nonlinear ordinary differential equation

Q. $\frac{d^{2}y}{d{x}^2}+(x^2+4x)\frac{dy}{dx}+y=x^8-8$

The above equation is a

1. partial differential equation
2. nonlinear differential equation
3. non-homogeneous differential equation
4. ordinary differential equation

Q. The differential equation
$y^{\prime \prime }+(y^{3}sinx{)}^5{y}^{\prime }+y=cos{x}^3$ is

1. homogeneous
2. nonlinear
3. second order linear
4. nonhomogeneous with constant coeffecients

Q. The following differential equation has $3\left(\frac{d^{2}y}{d{t}^2}\right)+4{\left(\frac{dy}{dt}\right)}^3+y^2+2=x$

1. degree = 2, order = 1
2. degree = 1, order = 2
3. degree = 4, order = 3
4. degree = 2, order = 3

Q. The differential equation $EI\frac{d^{4}y}{d{x}^4}+P\frac{d^{2}y}{d{x}^2}+ky=0$ is

(where E, I, P, K are functions of x -only)

1. Linear of Fourth order
2. Non-linear of Fourth order
3. Non-Homogeneous
4. Linear and Fourth degree