Differential Equations Definition

Q. Show that the differential equation $(x^2+xy)dy=(x^2+y^2)dx$ is homogeneous and solve it.

Q.

The order of the differential equation of all circles of radius $r$, having a centre on the $y$-axis and passing through the origin is

1. $1$

2. $2$

3. $3$

4. $4$

Q.

A differential equation of first order and first degree is

1. 
   

2. 
   

3. 
   

4. None of these

Q. The solution of the differential equation $x\frac{dy}{dx}-y=3$ represents a family of

1. straight lines
2. circles
3. parabolas
4. ellipses

Q.

If the equation $a{x}^2+b{y}^2+cx+cy=0$, represents a pair of straight lines, then

1. $a(b+c)=0$

2. $b(c+a)=0$

3. $c(a+b)=0$

4. $a+b+c=0$

Q.

Which of the following differential equations has y=x as one of its particular solution?
(a) $\frac{d^{2}y}{d{x}^2}-x^2\frac{dy}{dx}+xy=x$
(b) $\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+xy=x$
(c) $\frac{d^{2}y}{d{x}^2}-x^2\frac{dy}{dx}+xy=0$
(d) $\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+xy=0$

Q. Show that the differential equation
$(1+e^{\frac{x}{y}})dx+e^{\frac{x}{y}}(1-\frac{x}{y})dy=0$
is homogenous and solve it.

Q. Find the general solution of the differential equation $\frac{dy}{dx}+3y=e^{-2x}$

Q.

The equation of the circle of radius $3$ that lies in the fourth quadrant and touches the lines $x=0$ and $y=0$is:

1. $x^2+y^2-6x+6y+9=0$

2. $x^2+y^2-6x-6y+9=0$

3. $x^2+y^2+6x-6y+9=0$

4. $x^2+y^2+6x+6y+9=0$

Q.

Let Z = ax + by is a linear objective function. Variables x and y are called ……… variables.

1. Independent

2. Decision

3. Continuous

4. Dependent

Q. Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
$y=e^x+1:y"-y^{\prime }=0$

Q. Solve the differential equation : $(x+1)\frac{dy}{dx}=2e^{-y}-1;y\left(0\right)=0.$

Q. Find the order and degree of the following differential equation $\frac{d^{3}y}{d{x}^3}+{\left(\frac{d^{2}y}{d{x}^2}\right)}^3+\frac{dy}{dx}=xy$

Q. A relation in dependent variables and independent variables satisfying a differential equation is called general solution of that differential equation.

1. True
2. False

Q.

Identify the type of differential equation

1. Variable separable

2. Homogeneous

3. Linear

4. Both Variable separable and Linear

Q. The general solution of the differential equation $\frac{dy}{dx}=e^{x+y}\text{is}$

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

Q.

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(i)

(ii)

(iii)

(iv)

Q.

Find the equation of a curve passing through the point (0, 0) and whose differential equation is.

Q. General solution of $y\frac{dy}{dx}+b{y}^2=a\cos x,0<x<1$ is

1. $y^2=2a(2b\sin x+\cos x)+c{e}^{-2bx}$
2. $(4b^2+1)y^2=2a(\sin x+2b\cos x)+c{e}^{-2bx}$
3. $(4b^2+1)y^2=2a(\sin x+2b\cos x)+c{e}^{2bx}$
4. $y^2=2a(2b\sin x+\cos x)+c{e}^{-2bx}$

Q. Find the general solution of the differential equation $(x+3y^2)\frac{dy}{dx}=y(y>0)$

Q. Find the general solution of the differential equation $x\frac{dy}{dx}+2y=x^2\text{log}x$

Q.

The velocity of a particle v at any instant is

$\stackrel{\rightharpoonup }{v}=y\stackrel{\rightharpoonup }{i}+x\stackrel{\rightharpoonup }{j}$

The equation of trajectory of the particle is:

1. $x^2+y^2=cons\tan t$

2. $y^2=x^2+cons\tan t$

3. xy=constant

4. None of these

Q.

Find a particular solution of the differential equation $\frac{dy}{dx}+ycotx=4xcosecx(x\ne 0)$ given that y=0 when $x=\frac{\pi }{2}$

Q. Solve the differential equation $\frac{dy}{dx}+2xy=y$

Q. The differential equation whose general solution is given by, $y=\left(c_1\cos \right(x+c_2\left)\right)-\left(c_3{e}^{(-x+c_4)}\right)+\left(c_5\sin x\right),$ where $c_1,c_2,c_3,c_4,c_5$ are constants, is

1. $\frac{d^{4}y}{d{x}^4}-\frac{d^{2}y}{d{x}^2}+y=0$
2. $\frac{d^{3}y}{d{x}^3}+\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$
3. $\frac{d^{5}y}{d{x}^5}+y=0$
4. $\frac{d^{3}y}{d{x}^3}-\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$

Q. Find the general solution of the differential equation $\frac{dy}{dx}+\frac{y}{x}=x^2$

Q.

Determine order and degree (when defined) of differential equations.
$\frac{d^{4}y}{d{x}^4}+sin\left(y^{\prime \prime \prime }\right)=0$

Q. Which of the following are differential equations ?

1. $\frac{dz}{dy}=z$
2. $\frac{dz}{dy}\times z=0$
3. $x^2+y^2+2xy+2x-3y+25=0$
4. $w^2+d^2+\frac{dw}{dx}+5=0$

Q. For the differential equation find the general solution of
$x^5\frac{dy}{dx}=-y^5$

Q.

Show that the given differential equation is homogeneous and then solve it.

$x\frac{dy}{dx}-y+xsin\left(\frac{y}{x}\right)=0$