Differential Equations

Subtopic: Higher Order Linear ODEs with Constant Coefficients

Consider the differential equation with the initial conditions given below. If y(x) is the solution of the equation, the value of the slope, $\frac{dy}{dx}$ , at x = ln(2) is ________ (rounded off to three decimal places). $\frac{d^2y}{dx^2}$ + 2 $\frac{dy}{dx}$ + y = 0 with y|x=0 = 0 and $\frac{dy}{dx}$| x=0 = 1

Subtopic: Partial Differential Equations

The partial differential equation $\frac{\partial^2 u}{\partial x^2} + 4\frac{\partial^2 u}{\partial x\partial y} + 2\frac{\partial^2 u}{\partial y^2} = 0$ is:

Subtopic: First Order ODEs

Consider the ordinary differential equation: $\frac{1}{2} \frac{dy}{dx} + \frac{y}{x} = 1$. If $y = 2/3$ at $x = 1$, then the value of $y$ at $x = 3$ is ______ (rounded off to the nearest integer).

Subtopic: First Order ODEs

The acceleration of a body travelling in a straight line is given by $a = -C_1 - C_2 v^2$, where $v$ is the velocity, and $C_1, C_2$ are positive constants. Starting with an initial positive velocity $v_0$, the distance travelled by the body before coming to rest for the first time is:

Subtopic: First Order ODEs

At a point in the trajectory of an unpowered space vehicle moving about the Earth, the altitude above the mean sea level is 600 km , and the speed with reference to a coordinate system fixed to the center of mass of the Earth is 9 km/s. Assume that the Earth is a sphere with a radius 6400 km and GMEartℎ = 3.98 × 1014 m^3/s^2, where, G is the universal gravitational constant and MEartℎ is mass of the Earth. The trajectory is:

Subtopic: First Order ODEs

Consider the differential equation 2 2 2 4 2 0d y dyx x ydx dx+ + = for 1x  with initial conditions 0y = , 1$\frac{dy}{dx}$ = at 1x = . The value of y at 2x = is ____ . (round off to two decimal places)

Subtopic: First Order ODEs

Consider the differential equation $\frac{d^2 y}{dx^2} - 2 \frac{dy}{dx} + y = 0$. The boundary conditions are $y = 0$ and $\frac{dy}{dx} = 1$ at $x = 0$. Then the value of $y$ at $x = 1/2$ is:

Subtopic: Partial Differential Equations

Consider the partial differential equation $\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0$ where $x, y$ are real. If $f(x, y) = a(x)b(y)$, where $a(x)$ and $b(y)$ are real functions, which one of the following statements can be true?

Subtopic: First Order ODEs

A cylindrical object of diameter $900\text{ mm}$ is designed to move axially in air at $60\text{ m/s}$. Its drag is estimated on a geometrically half-scaled model in water. Coefficients of dynamic viscosity and densities for air and water are $1.86 \times 10^{-5}\text{ Pa}\cdot\text{s}, 1.2\text{ kg/m}^3$ and $1.01 \times 10^{-3}\text{ Pa}\cdot\text{s}, 1000\text{ kg/m}^3$ respectively. Drag measured for the model is $2280\text{ N}$. Drag experienced by the full-scale object is _____ $\text{N}$ (rounded off to the nearest integer).

Subtopic: First Order ODEs

Consider the differential equation $\frac{d^2 y}{dx^2} + 8\frac{dy}{dx} + 16y = 0$ with boundary conditions $y(0) = 1$ and $\frac{dy}{dx}(0) = 0$. The solution to this equation is:

Subtopic: Higher Order Linear ODEs with Constant Coefficients

In an aerospace engineering analysis of Differential Equations (Higher Order Linear ODEs with Constant Coefficients), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?

Subtopic: First Order ODEs

The function $y(x) = A e^{mx} + B e^{-mx}$, where $A, B$ and $m$ are constants, is a solution of which differential equation?

Subtopic: Partial Differential Equations

The similarity solution for the diffusion equation $\frac{\partial u}{\partial t} = \nu \frac{\partial^2 u}{\partial y^2}$ is $u(y, t) = u(\eta)$, where the similarity variable $\eta$ is proportional to:

Subtopic: First Order ODEs

The differential equation $x \frac{dy}{dx} + y = c$, where $c$ is a constant, represents a family of:

Subtopic: Higher Order Linear ODEs with Constant Coefficients

In an aerospace engineering analysis of Differential Equations (Higher Order Linear ODEs with Constant Coefficients), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?

Subtopic: First Order ODEs

The curve $y = f(x)$ is such that its slope is equal to $y^2$ for all real $x$. If the curve passes through $(1, -1)$, the value of $y$ at $x = 0$ is _________ (round off to 1 decimal place).

Subtopic: First Order ODEs

Assuming ISA standard sea level conditions (288.16 K, density of 1.225 kg/m^3, g = 9.81 m/s^2, R = 287 J/(kg-K)), the density (in kg/m^3) of air at Leh, which is at an altitude of 3500 m above mean sea level is ________ (accurate to two decimal places).

Subtopic: First Order ODEs

Consider a cubical tank of side 2 m with its top open. It is filled with water up to a height of 1 m. Assuming the density of water to be 1000 kg/m^3, g as 9.81 m/s2 and the atmospheric pressure to be 100 kPa, the net hydrostatic force (in kN) on the side face of the tank due to the air and water is ______ (accurate to two decimal places).

Subtopic: First Order ODEs

The effective stiffness of the spring-mass system as shown in the figure below is ____ N/mm.

Subtopic: Partial Differential Equations

The partial differential equation 2 2 x u t u     , where $\alpha$ is a positive constant, is

Subtopic: Higher Order Linear ODEs with Constant Coefficients

Consider a second order linear ordinary differential equation 0 y 4dx dy4 dx y d 2 2   , with the boundary conditions 1dx dy; 1 ) 0 ( y 0 x   . The value of y at x =1 is

Subtopic: First Order ODEs

The governing differential equation of motion of a damped system is given by 0 kxdt dxcdt x dm 2 2   . If m = 1 kg, c = 2 Ns/m and k = 2 N/m then the frequency of the damped oscillation of this system is ____ rad/s.

Subtopic: Higher Order Linear ODEs with Constant Coefficients

In an aerospace engineering analysis of Differential Equations (Higher Order Linear ODEs with Constant Coefficients), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?

Subtopic: First Order ODEs

A damped single degree of freedom system whose undamped natural frequency,𝝎𝝎𝒏𝒏=10Hz, is subjected to sinusoidal external force. Power is half of the maximumfor the two frequencies of 60.9469 rad/s and 64.7168 rad/s. The damping factor associated with the vibrating system (in %) is _______.

Subtopic: First Order ODEs

A 1.8m long steel beam of rectangular cross section (10mm× 6mm) is simply supported with a length of 1.2m between the supports and an overhang of 0.3m on either side. Young's modulus for the material of the beam is 200GPa. For a 50N load applied at the center of the beam, magnitude of the slopeof the beam at tip S is _______.

Subtopic: First Order ODEs

A function ሻ satisfies the differential equation 2 2 20dy d y ydt dt  and is subject to the initial conditions (0 ) 0yt  and (0 ) 1dy tdt   . The value of (1 )yt  is

Subtopic: First Order ODEs

If ()ut is a unit step function, the solution of the differential equation 2 2 ()dxmk x u tdt += in Laplace domain is

Subtopic: First Order ODEs

The general solution of the differential equation 20dy ydx- = is

Subtopic: Partial Differential Equations

The nth derivative of the function 1 3y x= + is

Subtopic: First Order ODEs

Ihrust produced by this engine under the given conditions?

Subtopic: First Order ODEs

After 10 seconds, the change in specific internal energy of the air is

Subtopic: Higher Order Linear ODEs with Constant Coefficients

If the combustion takes place with the fuel to air ratio 0.028, then the equivalence ratio of the fuel-oxidizer mixture is

Subtopic: First Order ODEs

Let a dynamical system be described by the differential equation $\frac{d^2x}{dt^2} + \cos x = 0$. Which of the following differential equations describes this system in a close approximation sense for small perturbation $\Delta x$ about $x = \pi/4$?