Quiz about PDE First and Higher Order Quiz

Question 1

Find the general solution of the linear partial differential equation, yzp+zxq=xy.

Accepted Answer

$$\phi\left(x^2-y^2,y^2-z^2\right)=0$$

Answer

$$\phi\left(x^2-z^2,\ y^2-z^2\right)=0$$

Answer

$$\phi\left(x^2-y^2,-z^2\right)=0$$

Answer

$$\phi\left(x^2-y^2,y^2-x^2\right)=0$$

Question 2

The solution of Lagrange's partial differential equation xp+yq=z is

Accepted Answer

$$f\left(\frac{x}{y},\frac{y}{z}\right)=0$$

Answer

$$f\left(\frac{x}{y},\frac{x}{z}\right)=0$$

Answer

$$f\left(\frac{y}{x},\frac{y}{z}\right)=0$$

Answer

$$f\left(\frac{y}{x},\frac{x}{z}\right)=0$$

Question 3

The solution of non-linear partial differential equation

$$\sqrt{p}+\sqrt{q}=1$$ is

Accepted Answer

$$z=ax+\left(1-\sqrt{a}\right)^2\ y+c$$

Answer

$$z=ax+ay+b$$

Answer

$$z=ax+2ay+c$$

Answer

$$z=\sqrt{a}x+ay+c$$

Question 4

The solution of non-linear partial differential equation

$$\sqrt{p}+\sqrt{q}=x$$ is

Accepted Answer

$$z=\frac{\left(x-a\right)^3}{3}+a^2y+b$$

Answer

$$z=\left(x-a\right)^3+a^2y+b$$

Answer

$$z=\left(x-a\right)^3+b$$

Answer

$$z=a^2x+\left(y-a\right)^3+c$$

Question 5

The solution of non-linear partial differential equation

$$p\left(1+q\right)=qz$$ is

Accepted Answer

$$\log\left(az-1\right)=x+a\ y+c$$

Answer

$$\log z=x+a\ y+c$$

Answer

$$az-1=e^{x+a\ y+c}$$

Question 6

The solution of the partial differential equation

$$q-p+x-y=0$$ is

Accepted Answer

$$z=\frac{\left(x+a\right)^2}{2}+\frac{\left(y+a\right)^2}{2}+c$$

Answer

$$z=\left(x+a\right)^2+\left(y+a\right)^2+c$$

Answer

$$z=\frac{\left(x+a\right)^2}{2}+\frac{\left(y-a\right)^2}{2}+c$$

Answer

$$z=\frac{\left(x-a\right)^2}{2}+\frac{\left(y+a\right)^2}{2}+c$$

Question 7

An equation containing partial derivatives of one or more dependent variables of two or more independent variables

Accepted Answer

Partial Differential Equation

Answer

Ordinary Partial Differential Equation

Answer

Ordinary Differential Equation

Answer

Partially Ordinary Differential Equation

Question 8

Find the differential equation of all spheres of fixed radius having centre in xy -plane. hint:

$$\left(x-a\right)^2+\left(y-b\right)^2+z^2=r^2$$

Accepted Answer

$$z^2\left(p^2+q^2+1\right)=r^2$$

Answer

$$z^2\left(p^2+q^2\right)=r^2$$

Answer

$$p^2+q^2+1=r^2$$

Question 9

What is the PDE when the arbitrary constants a and b are eliminated from

$$z=\left(x-a\right)^2+\left(y-b\right)^2$$

Accepted Answer

$$p^2+q^2=4z$$

Answer

$$p^2-q^2-4z=0$$

Answer

$$p^2q^2=4z$$

Answer

None

Question 10

What is the PDE when the arbitrary constants a and b are eliminated from

$$ax^2+by^2=1-z^2$$

Accepted Answer

$$zxp+zyq=z^2-1$$

Answer

$$zxp+zyq=z^2+1$$

Answer

None

Question 11

What is the PDE when the arbitrary constants a and b are eliminated from

z=axy+b

Accepted Answer

$$xp-yq=0$$

Answer

$$xp+yq=1$$

Answer

None

Question 12

Obtain partial differential equation by eliminating the arbitrary function from the given equation.

$$z=f\left(x^2+y\right)+g\left(x^2-y\right)$$

Accepted Answer

$$rx-p=4x^3t$$

Answer

$$rx-t=4x^3p$$

Answer

$$rx-q=4t$$

Answer

none

Question 13

obtain PDE by eliminating arbitrary functions from the given equation

$$z=\phi\left(x^2-y^2\right)$$

Accepted Answer

py+qx=0

Answer

px+qy=0

Answer

p/x+q/y=9

Answer

x/p+y/q=0

Question 14

The solution of Lagrange's partial differential equation xp+yq=z

Accepted Answer

$$f\left(\frac{x}{y},\frac{y}{z}\right)$$

Answer

$$f\left(\frac{x}{y},\frac{x}{z}\right)$$

Answer

$$f\left(\frac{x}{z},\frac{y}{z}\right)$$

Answer

$$f\left(\frac{y}{x},\frac{y}{z}\right)$$

Question 15

The solution of Lagrange's partial differential equation xp+zq=y

Accepted Answer

$$f\left(y^2-z^2,\frac{y}{x+z}\right)$$

Answer

$$f\left(y^2-z^2,\frac{x}{y+z}\right)$$

Question 16

The solution of Lagrange's partial differential equation $$\left(1-x\right)p+\left(2-y\right)q=3-z$$

Accepted Answer

$$f\left(\frac{2-y}{1-x},\frac{3-z}{2-y}\right)$$

Answer

$$f\left(\frac{3-y}{2-x},\frac{1-z}{2-y}\right)$$

Answer

$$f\left(\frac{2-x}{1-z},\frac{3-x}{2-y}\right)$$

Question 17

$$\ Solve:.\left(D^2-6DD$$

Accepted Answer

$$z=f_1(y+3x)+xf_2(y+3x)$$

Answer

$$z=f_1(y+3x)+yf_2(y+3x)$$

Answer

$$z=f_1(y-3x)+xf_2(y-3x)$$

Answer

$$z=f_1(y-3x)+yf_2(y-3x)$$

Question 18

$$.$$ Find the P.I of $$\left(D^3+D^2D$$

Accepted Answer

$$0$$

Answer

$$1$$

Answer

$$e^x$$

Answer

$$none$$

Question 19

$$.$$ Find the P.I of $$\left(D^2-5DD$$

Accepted Answer

$$\frac{1}{3}e^{\left(x+y\right)}$$

Answer

$$e^{\left(x+y\right)}$$

Answer

$$\frac{1}{2}e^{\left(x+y\right)}$$

Answer

$$\frac{1}{4}e^{\left(x+y\right)}$$

Question 20

$$\ .$$ Find the P.I of $$\left(D^3-3D^2D$$

Accepted Answer

$$\ \frac{1}{27}e^{\left(x+2y\right)}$$

Answer

$$\ \frac{1}{6}e^{\left(x+2y\right)}$$

Answer

$$\ \frac{1}{27}e^{\left(2x+y\right)}$$

Answer

$$\ \frac{1}{6}e^{\left(2x+y\right)}$$

Question 21

$$.\ $$ Find the A.E of $$\left(D^4-D$$

Accepted Answer

$$1,-1,+i,-i$$

Answer

$$1,1,+i,-i$$

Answer

$$1,1,+i,+i$$

Answer

None

Question 22

$$\ $$ Find the A.E of $$\left(D^4-2D^3D$$

Accepted Answer

$$m^4-2m^3+2m+1=0$$

Answer

$$m^4+2m^3+2m+1=0$$

Answer

$$m^4-2m^2-2m+1=0$$

Answer

$$m^4-2m^3+2m^2+1=0$$

Question 23

$$.$$ Find the P.I of $$\left(D^2-2DD$$

Accepted Answer

$$-\frac{1}{16}\cos(x-3y)$$

Answer

$$-\frac{1}{16}\sin⁡(x-3y)$$

Answer

$$\frac{1}{16}\cos⁡(x-3y)$$

Answer

$$\frac{1}{16}\sin⁡(x-3y)$$

Question 24

$$.$$ Find the P.I of $$\left(D^2-4DD$$

Accepted Answer

$$\frac{x^2}{2}e^{\left(2x+y\right)}$$

Answer

$$\ \frac{y^2}{2}e^{\left(2x+y\right)}$$

Answer

$$\frac{xy^{ }}{2}e^{\left(2x+y\right)}$$

Answer

$$\frac{x^2}{2}e^{\left(2y+x\right)}$$

Question 25

$$.$$ Find the P.I of $$\left(D^2+3DD$$

Accepted Answer

$$\frac{1}{4}\sin⁡y$$

Answer

$$\frac{1}{4}\cos⁡y$$

Answer

$$\frac{1}{4}\cos⁡x$$

Answer

$$\frac{1}{4}\sin x$$

Question 26

$$.$$ What is the order of the $$\left(\frac{\partial^2u}{\partial x^2}\right)^3+\left(\frac{\partial u}{\partial y}\right)^4=0$$ PDE

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 27

Find the complete integral of pq=1 .

Accepted Answer

(i)&(ii)

Answer

i) z=(1/b)x+by+c

Answer

ii) z=ax+(1/a)y+c

Answer

iii) z=ax+by+c

Question 28

Find the singular integral of z=px+qy+pq

Accepted Answer

z+xy=0

Answer

z=x+y+1

Answer

xy-z=0

Answer

zx+y=0

Question 29

Solve: p+q=1

Accepted Answer

both (i)&(ii)

Answer

i) f(x-y,y-z)=0

Answer

ii) f(x-y, x-z)=0

Answer

none

Question 30

What are the multiples while solving $$\left(3z-4y\right)p+\left(4x-2z\right)q=2y-3x$$

Accepted Answer

$$2,3,4$$

Answer

$$3,4,2$$

Answer

$$4,2,3$$

Answer

$$1,1,1$$

Question 31

For which type of the PDE the singular integral does exist

Accepted Answer

$$z=px+qy+f\left(p,q\right)$$

Answer

$$F\left(p,q\right)=0$$

Answer

$$F\left(z,p,q\right)=0$$

Answer

none

Question 32

Accepted Answer

Option C is correct

Answer

A.

Answer

B.

Answer

D.

Question 33

Accepted Answer

Singular Integral

Answer

General Integral

Answer

Complete Integral

Answer

Particular Solution

Question 34

Accepted Answer

Option B is correct

Answer

A.

Answer

C.

Answer

D.

Question 35

Accepted Answer

Option B is correct

Answer

A.

Answer

C.

Answer

D.

Question 36

Accepted Answer

Option A is correct

Answer

B.

Answer

C.

Answer

D.

Question 37

Find the general solution of the linear partial differential equation, yzp+zxq=xy.

Accepted Answer

$$\phi\left(x^2-y^2,y^2-z^2\right)=0$$

Answer

$$\phi\left(x^2-z^2,\ y^2-z^2\right)=0$$

Answer

$$\phi\left(x^2-y^2,-z^2\right)=0$$

Answer

$$\phi\left(x^2-y^2,y^2-x^2\right)=0$$

Question 38

The solution of Lagrange's partial differential equation xp+yq=z is

Accepted Answer

$$f\left(\frac{x}{y},\frac{y}{z}\right)=0$$

Answer

$$f\left(\frac{x}{y},\frac{x}{z}\right)=0$$

Answer

$$f\left(\frac{y}{x},\frac{y}{z}\right)=0$$

Answer

$$f\left(\frac{y}{x},\frac{x}{z}\right)=0$$

Question 39

The solution of non-linear partial differential equation

$$\sqrt{p}+\sqrt{q}=1$$ is

Accepted Answer

$$z=ax+\left(1-\sqrt{a}\right)^2\ y+c$$

Answer

$$z=ax+ay+b$$

Answer

$$z=ax+2ay+c$$

Answer

$$z=\sqrt{a}x+ay+c$$

Question 40

The solution of non-linear partial differential equation

$$\sqrt{p}+\sqrt{q}=x$$ is

Accepted Answer

$$z=\frac{\left(x-a\right)^3}{3}+a^2y+b$$

Answer

$$z=\left(x-a\right)^3+a^2y+b$$

Answer

$$z=\left(x-a\right)^3+b$$

Answer

$$z=a^2x+\left(y-a\right)^3+c$$

Question 41

The solution of non-linear partial differential equation

$$p\left(1+q\right)=qz$$ is

Accepted Answer

$$\log\left(az-1\right)=x+a\ y+c$$

Answer

$$\log z=x+a\ y+c$$

Answer

$$az-1=e^{x+a\ y+c}$$

Question 42

The solution of non-linear partial differential equation

$$q-p+x-y=0$$ is

Accepted Answer

$$z=\frac{\left(x+a\right)^2}{2}+\frac{\left(y+a\right)^2}{2}+c$$

Answer

$$z=\left(x+a\right)^2+\left(y+a\right)^2+c$$

Answer

$$z=\frac{\left(x+a\right)^2}{2}+\frac{\left(y-a\right)^2}{2}+c$$

Answer

$$z=\frac{\left(x-a\right)^2}{2}+\frac{\left(y+a\right)^2}{2}+c$$

Question 43

$$\int_0^{2\pi}\ x\ \cos nx\ dx\ =$$

Accepted Answer

$$\left|x\left(\frac{\sin nx}{n}\right)-\left(-\frac{\cos nx}{n^2}\right)\right|_0^{2\pi}\ \ =0$$

Answer

$$\left|x\left(\sin x\right)\left(-\cos x\right)\right|$$

Answer

$$\left|x\left(\frac{\sin nx}{n}\right)-\left(-\frac{\cos nx}{n^2}\right)\right|$$

Answer

$$\left|x\left(\frac{\sin nx}{n}\right)-\left(-\frac{\cos nx}{n^2}\right)\right|_0^{2\pi}\ =2$$

Question 44

$$\int_0^{2\pi}\ \ x\ \sin x\ dx$$ =

Accepted Answer

$$\left|x\left(-\cos x\right)-\left(-\sin x\right)\right|_0^{2\pi}\ =\ -2\pi$$

Answer

$$\left|x\left(-\cos x\right)-\left(-\sin x\right)\right|$$

Answer

$$\left|x\left(\cos x\right)-\left(\sin x\right)\right|$$

Answer

$$\left|x\left(-\cos x\right)-\left(-\sin x\right)\right|_0^{2\pi}\ =\ 0$$

Question 45

The value of cos $$n\pi=$$ ------ and Cos 0=-------

Accepted Answer

$$\left(-1\right)^n$$ and 1

Answer

0 and 0

Answer

$$\left(-1\right)^n$$ and 0

Answer

1 and 1

Question 46

A PDE consists of

Accepted Answer

more than one independent variable

Answer

only one independent variable

Answer

more than one dependent varible

Question 47

The highest derivative in the given pde is called

Accepted Answer

order

Answer

degree

Question 48

If the number of arbitrary constants is less than or equal to number of independent variables then the pde obtained is of order

Accepted Answer

=1

Answer

>1

Answer

<1

Question 49

If the number of arbitrary function is n then the pde obtained is of order

Accepted Answer

=n

Answer

<n

Answer

>n

Answer

1

Question 50

The complete integral is defined as

Accepted Answer

The number of arbitrary constants = The number of independent variables

Answer

The number of arbitrary constants < The number of independent variables

Answer

The number of arbitrary constants > The number of independent variables

Question 51

Giving particular values for arbitrary constants in the complete integral is called

Accepted Answer

Particular solution

Answer

General solution

Answer

Singular solution

Question 52

If we eliminate a and b from z = (x+a)(y+b)

Accepted Answer

z = pq

Answer

z = p/q

Answer

z = p - q

Answer

z = p + q

Question 53

If the number of constants to be eliminated is equal to number of independent variables, then the result is_____

Accepted Answer

First order PDE

Answer

Second order PDE

Answer

third order PDE

Answer

Second order linear PDE

Question 54

If the number of constants to be eliminated is greater than the number of independent variables, then the result is_____

Accepted Answer

Second and higher order PDE

Answer

First order PDE

Answer

only Second order linear PDE

Question 55

If the equation is in the form f(x, p, q)=0, then we assume

Accepted Answer

q=a

Answer

p=a

Answer

z=pq

Answer

p=aq

Question 56

The solution of the equation z = px + qy + pq is____

Accepted Answer

z = ax + by + ab

Answer

z = ax + by + pq

Answer

z = bx + ay + ab

Answer

z = px + qy + ab

Question 57

Eliminate the arbitrary function from z = f(x²+y²)

Accepted Answer

py = qx

Answer

px = qy

Answer

py+qx=0

Answer

px+qy=0

Question 58

Eliminate the arbitrary constants c and r from x²+y²+(z-c)²=r²

Accepted Answer

py = qx

Answer

px = qy

Answer

px+qy=0

Answer

py+qx=0

Question 59

A solution containing as many arbitrary constants as there are independent variables is called____

Accepted Answer

complete integral

Answer

particular integral

Answer

singular integral

Answer

general integral

Question 60

A solution obtained by giving particular values to the arbitrary constants in a complete integral is called____

Accepted Answer

particular integral

Answer

complete integral

Answer

singular integral

Answer

general integral

Question 61

A partial differential equation which is not linear then it is called_____

Accepted Answer

non-linear

Answer

semi linear

Answer

collinear

Answer

Quasi linear

Question 62

A partial differential equation requires

Accepted Answer

two or more independent variables

Answer

exactly one independent variable

Answer

more than one dependent variable

Answer

equal number of dependent and independent variables

Question 63

Form a PDE by eliminating arbitrary constants from z= ax + by

Accepted Answer

z = px + qy

Answer

z = px - qy

Answer

z = qx + py

Answer

z = qx - py

Question 64

Form a PDE by eliminating arbitrary constants from z= a(x+y) + b

Accepted Answer

p-q=0

Answer

p+q=0

Answer

px = qy

Answer

qx = py

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