Homogeneous Linear Differential Equations (General Form of LDE)

Q.

Explain the procedure to find $\frac{d{}^{2}y}{d{x}^2}$ [second order derivative] of any function y=f(x).

Q. The general solution of the differential equation $(D^2-4D+4)y=0$ is of the form $(givenD=\frac{d}{dx}and{C}_1,C_{2}areconstants)$

1. $C_1{e}^{2x}$
2. $C_1{e}^{2x}+C_2{e}^{-2x}$
3. $C_1{e}^{2x}+C_2{e}^{2x}$
4. $C_1{e}^{2x}+C_{2}x{e}^{2x}$

Q. The solution to the ordinary differential equation $\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}-6y=0$ is

1. $y=C_1{e}^{3x}+C_2{e}^{-2x}$
2. $y=C_1{e}^{3x}+C_2{e}^{2x}$
3. $y=C_1{e}^{-3x}+C_2{e}^{2x}$
4. $y=C_1{e}^{-3x}+C_2{e}^{-2x}$

Q.

If $\mathrm{y}=\mathrm{y}\left(\mathrm{x}\right)$ is the solution of the differential equation, $\frac{\mathrm{dy}}{\mathrm{dx}}+2\mathrm{ytan}\mathrm{x}=\sin \mathrm{x},\mathrm{y}\left(\frac{\mathrm{\pi }}{3}\right)=0$, then the maximum value of the function $\mathrm{y}\left(\mathrm{x}\right)$ over $\mathrm{R}$ is equal to

1. $8$

2. $\frac{1}{2}$

3. $-\frac{15}{4}$

4. $\frac{1}{8}$

Q. The solution for the following differential equation with boundary conditions $y\left(0\right)=2$ and $y^{\prime }\left(1\right)=-3$ is, $\frac{d^{2}y}{d{x}^2}=3x-2$

1. $y=\frac{x^3}{3}-\frac{x^2}{2}+3x-6$
2. $y=3x^3-\frac{x^2}{2}-5x+2$
3. $y=\frac{x^3}{2}-x^2-\frac{5x}{2}+2$
4. $y=x^3-\frac{x^2}{2}+5x+\frac{3}{2}$

Q. The solution of the differential equation $\frac{d^{2}y}{d{t}^2}+2\frac{dy}{dt}+y=0$ with $y\left(0\right)=y^{\prime }\left(0\right)=1$ is

1. $(2-t)e^t$
2. $(1+2t)e^{-t}$
3. $(2+t)e^{-t}$
4. $(1-2t)e^t$

Q. $y=e^{-2x}$ is a solution of the differential equation $y^{\prime \prime }+y^{\prime }-2y=0$

1. True
2. False

Q.

If $\omega =e^{\frac{i2\pi }{3}}$ and $a,b,c,x,y,z$ are non-zero complex numbers such that $a+b+c=x,a+b\omega +c{\omega }^2=y,a+b{\omega }^2+c\omega =z$. Then, the value of $\frac{\left(\right|x{|}^2+|y{|}^2+|z{|}^2)}{\left(\right|a{|}^2+|b{|}^2+|c{|}^2)}$.

1. $2$

2. $3$

3. $1$

4. $4$

Q. If $\frac{d^{2}y}{d{t}^2}+y=0$ under the conditions $y=1,\frac{dy}{dt}=0$, when t=0 then y is equal to

1. $sint$
2. $cost$
3. $tant$
4. $cott$

Q. The complete solution of the ordinary differential equation $\frac{d^{2}y}{d{x}^2}+p\frac{dy}{dx}+qy=0$ is $y=c_1{e}^{-x}+c_2{e}^{-3x}$


Then, $p$ and $q$ are

1. $p=3$, $q=3$
2. $p=3,q=4$
3. $p=4$, $q=3$
4. $p=4$, $q=4$

Q.

If $y=f\left(x^2+2\right)$ and $f^{}\left(3\right)=5$, then $\frac{dy}{dx}$at $x=1$ is

1. $5$

2. $25$

3. $15$

4. $10$

Q. The particular solution of the initial value problem given below is $\frac{d^{2}y}{d{x}^2}+12\frac{dy}{dx}+36y=0$ with $y\left(0\right)=3$ and ${\begin{array}{c}\frac{dy}{dx}\end{array}|}_{x=0}=-36$

1. $(3-18x)e^{-6x}$
2. $(3+25x)e^{-6x}$
3. $(3+20x)e^{-6x}$
4. $(3-12x)e^{-6x}$

Q. The differential equation $\frac{d^{2}y}{d{x}^2}+16y=0$ for $y\left(x\right)$ with the two boundary conditions ${\begin{array}{c}\frac{dy}{dx}\end{array}|}_{x=0}=1$ and ${\begin{array}{c}\frac{dy}{dx}\end{array}|}_{x=\frac{1}{2}}=-1$ has

1. no solution
2. exactly two solutions
3. exactly one solution
4. infinitely many solutions

Q. The solution for the differential equation $\frac{d^{2}x}{d{t}^2}=-9x$ with initial conditions $x\left(0\right)=1$ and ${\begin{array}{c}\frac{dx}{dt}\end{array}|}_{t=0}=1,$ is

1. $t^2+t+1$
2. $sin3t+\frac{1}{3}cos3t+\frac{2}{3}$
   
3. $\frac{1}{3}sin3t+cos3t$
4. $cos3t+t$

Q. Consider the differential equation $\frac{d^{2}x\left(t\right)}{d{t}^2}+3\frac{dx\left(t\right)}{dt}+2x\left(t\right)=0$
Given $x\left(0\right)=20$ and $x\left(1\right)=10/e$, where $e=2.718$, the value of x(2) is

Q.

The solution of the differential equation $\frac{dy}{dx}+\frac{y}{x}=\sin \left(x\right)$ is

1. $x\left(y+\cos \left(x\right)\right)=\sin \left(x\right)+c$

2. $x\left(y-\cos \left(x\right)\right)=\sin \left(x\right)+c$

3. $x\left(y\cos \left(x\right)\right)=\sin \left(x\right)+c$

4. $x\left(y-\cos \left(x\right)\right)=\cos \left(x\right)+c$

5. $\mathrm{x}\left(\mathrm{y}+\cos \left(\mathrm{x}\right)\right)=\cos \left(\mathrm{x}\right)+\mathrm{c}$

Q. If H(x, y) is a homogeneous function of degree n, then x$\frac{\partial H}{\partial x}+y\frac{\partial H}{\partial y}=nH$.

1. True
2. False

Q. For the differential equation
$\frac{d^{2}x}{d{t}^2}+6\frac{dx}{dt}+8x=0$ with initial conditions $x\left(0\right)=1$ and ${\begin{array}{c}\frac{dx}{dt}\end{array}|}_{t=0}=0)$, the solution is

1. $x\left(t\right)=2e^{-6t}-e^{-2t}$
2. $x\left(t\right)=2e^{-2t}-e^{-4t}$
3. $x\left(t\right)=-e^{-6t}+2e^{-4t}$
4. $a\left(t\right)=e^{-2t}+2e^{-4t}$

Q. The solution of $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+17y=0$; $y\left(0\right)=1$, $\frac{dy}{dx}\left(\frac{\pi }{4}\right)=0$ in the range $0<x<\frac{\pi }{4}$ is given by

1. $e^{-x}(cos4x+\frac{1}{4}sin4x)$
2. $e^x(cos4x-\frac{1}{4}sin4x)$
3. $e^{-4x}(cos4x-\frac{1}{4}sin4x)$
4. $e^{-4x}(cos4x-\frac{1}{4}sin4x)$

Q. The maximum value of the solution $y\left(t\right)$ of the differential equation $y\left(t\right)+\stackrel{..}{y}\left(t\right)=0$ with the initial conditions $\stackrel{.}{y}\left(0\right)=1$ and $y\left(0\right)=1,$ for $t\ge 0$ is

1. $1$
2. $2$
3. $\pi$
4. $\sqrt{2}$

Q. If f(x, y, z) = ${(x^2+y^2+z^2)}^{\frac{1}{2}}$ then $\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$ is equal to

1. zero
2. 1
3. 2
4. -3${(x^2+y^2+z^2)}^{\frac{5}{2}}$

Q.

How do you read $\frac{dy}{dx}?$

Q. Find the solution of $\frac{d^{2}y}{d{x}^2}=y$ which passes through origin and point $(ln2,\frac{3}{4})$

1. $y=\frac{1}{2}{e}^x-e^{-x}$
2. $y=\frac{1}{2}(e^x+e^{-x})$
3. $y=\frac{1}{2}(e^x-e^{-x})$
4. $y=\frac{1}{2}{e}^x+e^{-x}$

Q. The differential equation, $\frac{dy}{dx}+Py=Q$, is a linear equation of first order only if,

1. P is a constant but Q is a function of y
2. P and Q are functions of y or constants
3. P is a function of y but Q is a constant
4. P and Q are functions of x or constants

Q. For the equation $x^{\prime \prime }\left(t\right)+3x^{\prime }\left(t\right)+2x\left(t\right)=5$, the solution $x\left(t\right)$ approaches to the following values as $t\to \infty$

1. $0$
2. $5/2$
3. $5$
4. $10$

Q. The solution to the differential equation $\frac{d^{2}u}{d{x}^2}-k\frac{du}{dx}=0$ where k is constant, subjected to the boundary conditions $u\left(0\right)=0$ and $u\left(L\right)=U$, is

1. $u=U\frac{x}{L}$
2. $u=U\left(\frac{1-e^{kx}}{1-e^{kL}}\right)$
3. $u=U\left(\frac{1-e^{-kx}}{1-e^{-kL}}\right)$
4. $u=U\left(\frac{1+e^{kx}}{1+e^{kL}}\right)$

Q. It is given that $y^{\prime \prime }+2y^{\prime }+y=0,y\left(0\right)=0,y\left(1\right)=0$. What is $y\left(0.5\right)$?

1. $0$
2. $0.37$
3. $0.62$
4. $1.13$

Q. Which ONE of the following is a linear non- homogenous differential equation, where x and y are the independent and dependent variables respectively?

1. $\frac{dy}{dx}+xy=e^{-x}$
2. $\frac{dy}{dx}+xy=0$
3. $\frac{dy}{dx}+xy=e^{-y}$
4. $\frac{dy}{dx}+e^{-y}=0$

Q.

If $y^2=a{x}^2+bx+c$, where $a,b$ and $c$ are constant, then $y^3\frac{d^{2}y}{d{x}^2}$ is

1. a constant

2. a function of $x$

3. a function of $y$

4. a function of $x$ and $y$ both

Q. If the characteristic equation of the differential equation $\frac{d^{2}y}{d{x}^2}+2\alpha \frac{dy}{dx}+y=0$ has two equal roots, then the values of $\alpha$ are

1. $\pm 1$
2. $0,0$
3. $\pm j$
4. $\pm 1/2$