Free Printable Quiz on P. D. E 6th semester

1.

A partial differential equation requires

a)

exactly one independent variable

b)

two or more independent variables

c)

more than one dependent variable

d)

equal number of dependent and independent variables

2.

Using substitution, which of the following equations are solutions to the partial differential equation?

a)

b)

c)

d)

3.

The partial differential equation

is classified as

a)

elliptic

b)

parabolic

c)

hyperbolic

d)

none of the above

4.

The partial differential equation

is classified as

a)

elliptic

b)

parabolic

c)

hyperbolic

d)

none of the above

5.

The complete integral of $z=px+qy+f\left(p,q\right)$

a)

$z=px+qy$

b)

$z=ax+by+f\left(a,b\right)$

c)

$z=f\left(a,b\right)$

d)

none of the above

6.

What is the order & degree of the p.d.e $x^2\left(\frac{\partial z}{\partial x}\right)^2=z\left(x-y\frac{\partial z}{\partial y}\right)$

a)

1 & 2

b)

2 & 1

c)

2 & 2

d)

none of the above

7.

Singular solution of Partial differential equation can be obtained by

a)

General solution

b)

Complete solution

c)

both a & b

d)

none of the above

8.

The condition of compatibility of partial differential equations $f\left(x,y,z,p,q\right)=0\ ,\ \ g\left(x,y,z,p,q\right)=0$ is

a)

$\frac{\partial\left(f,g\right)}{\partial\left(p,q\right)}=0$

b)

$\frac{\partial\left(f,g\right)}{\partial\left(p,q\right)}\ne0$

c)

$\frac{\partial\left(f,p\right)}{\partial\left(g,q\right)}=0$

d)

$\frac{\partial\left(f,q\right)}{\partial\left(g,p\right)}\ne0$

9.

If u is homogeneous function of order n then $\frac{\partial u}{\partial x}\ ,\ \frac{\partial u}{\partial y}$ both are homogeneous function of order

a)

n

b)

n-1

c)

n+1

d)

n-2

10.

If $f\left(x,y\right)=0\$ then what is the value of $\frac{dy}{dx}$ is

a)

$\frac{f_x}{f_y}$

b)

$-\frac{f_x}{f_y}$

c)

$\frac{f_y}{f_x}$

d)

$-\frac{f_y}{f_x}$

11.

1. Find the nature of the one-dimensional wave equation.

a)

Hyperbolic

b)

Parabolic

c)

Elliptic

d)

None of These

12.

The nature of the one-dimensional heat equation is

a)

Circular

b)

Elliptic

c)

Parabolic

d)

Hyperbolic

13.

The nature of PDE 4u_xx+3 u_xy+3 u_yy=0

a)

Parabolic

b)

Hyperbolic

c)

Elliptic

d)

Laplace

14.

The PDE u_xx+ u_yy= 0, is known as

a)

1-D heat equation

b)

1-D wave equation

c)

Laplace equation

d)

None of these

15.

Let PDE u_xx+ u_yy= u_t, By method separation, we consider the solution

a)

u(x,y)=X(x)Y(y)

b)

u(x,y,t)=X(x)Y(y)T(t)

c)

u(x,t)=X(x)T(t)

d)

None of these

16.

Let PDE c²(u_xx+ u_yy)= u_t, By is known as

a)

2-D heat equation

b)

2-D wave equation

c)

Laplace equation

d)

None of these

17.

Let PDE c²(u_xx+ u_yy)= u_tt, By is known as

a)

2-D heat equation

b)

2-D wave equation

c)

Laplace equation

d)

None of these

18.

The nature of PDE 4u_xx+3 u_xy=0

a)

Circular

b)

Elliptic

c)

Hyperbolic

d)

Parabolic

19.

The nature of PDE u_xx+4 u_xy+3 u_yy,=0

a)

Circular

b)

Elliptic

c)

Hyperbolic

d)

Parabolic

20.

A rod of length 10 m has temperature 30⁰C and 40⁰C at end points. What is the temperature gradient

a)

1⁰C per cm

b)

3⁰C per cm

c)

2⁰C per cm

d)

None of these

21.

In the steady state, 2-D heat equation reduces to

a)

1-D heat equation

b)

1-D wave equation

c)

Laplace equation

d)

None of these

22.

In the steady state, the change in temperature at any point with respect to time is

a)

0⁰C

b)

>0⁰C

c)

<0⁰C

d)

None of these

23.

The ends A and B of a rod of length10cm are at 30⁰C and 80⁰C at end points until steady state prevails. Then Initial temperature distribution in the rod

a)

10+(5x)/2

b)

30+(5x)/2

c)

30+5x

d)

None of these

24.

The ends A and B of a rod of length20cm are at 30⁰C and 80⁰C at end points until steady state prevails. Then the temperature of the rod at ends are changed to 40⁰C and 60⁰C respectively. Final temperature distribution (i.e. in Steady state) is

a)

40+x

b)

50+x

c)

60+x

d)

70+x

25.

dy/dx = 4x/y. Suppose y(0)=1

The particular solution is

a)

B

b)

C

c)

D

d)

E

26.

The following Differential Equation is
$\frac{\text{d}y}{\text{d}x}=\frac{y}{x}$

a)

Separable.

b)

Non Separable

27.

Is this a Linear Differential equation

a)

Yes

b)

No

c)

Maybe

d)

red

28.

Solve the following differential equations:
$\frac{\text{d}y}{\text{d}x}=e^x$

a)

$1=e^x+C$

b)

$y=\frac{e^x}{2}+C$

c)

$y=e^x+C$

d)

$0=e^x+C$

29.

Which functions below will have the same slope field? (choose all that apply)

a)

y = x² - 3

b)

y = x² + 4

c)

y = 2x - 5

d)

y = 2x²

30.

In drawing the slope field for the differential equation , I would place short slope lines of _________ at the points (0,1), (1,-1), and (2,-2)
$\frac{dy}{dx}=2x-3y$

Select all that apply

a)

-3

b)

-1

c)

2

d)

5

e)

10

31.

y is proportional to the product of z and the square root of x

a)

$y=zx^2$

b)

$y=kzx^2$

c)

$y=kz\sqrt{x}$

d)

$y=zkx$

32.

The degree of a differential equation is define by a

a)

Positive real number

b)

Positive rational number

c)

Positive integer

d)

All the above

33.

Solution of a linear differential equation of first order can be obtain by

a)

multiplying on both the sides by its integrating factor.

b)

adding on both the sides by integrating factor.

c)

subtracting integrating factor from both sides

d)

None of the above

34.

The complete solution of Linear Differential equations involves

a)

complete function + particular integral

b)

complementary function + particular integral

c)

complementary function + definite integral

d)

complete function + indefinite integral

35.

The particular integral of the equation $\left(D-1\right)y=e^{3x}$ is

a)

$\frac{e^{3x}}{2}$

b)

$\frac{e^{-3x}}{2}$

c)

$\frac{e^{3x}}{4}$

d)

$\frac{e^x}{2}$

36.

The C.F. of the equation $\left(D^2-9\right)y=e^{-3x}+1+e^{3x}$ is

a)

$c_1e^{-3x}+c_2e^{-x}$

b)

$c_1e^{3x}+c_2e^{3x}$

c)

$c_1e^{3x}+c_2e^{-x}$

d)

$c_1e^{3x}+c_2e^{-3x}$

37.

Choose the right description for the given differential equation

a)

Ordinary, 2nd order, degree 2, non linear

b)

Partial, 1st order, degree 2, linear.

c)

Ordinary, 1st order, degree 2, linear

d)

Partial, 2nd order, degree 1, non linear

38.

Number of arbitrary constant in the general solution of a differential equation of degree 3 and order 4 is

a)

3

b)

4

c)

0

d)

4³

39.

$\frac{\text{d}^2x}{\text{d}y^2}+6\frac{\text{d}x}{\text{d}y}-5y=0\$

The general solution to the DE is,

a)

$y=Ae^x+Be^{5x}$

b)

$y=Ae^{-x}+Be^{-5x}$

c)

$y=Ae^{\left(-3+\sqrt{14}\right)x}+Be^{\left(-3-\sqrt{14}\right)x}$

d)

$y=A\cos\left(-3+\sqrt{14}\right)x+B\sin\left(-3-\sqrt{14}\right)x$

40.

Find a solution for y if dy/dx = 2x√y and y = 4 when x = 3.

a)

2√y = x² + C

b)

y = (x² + 25)²/4

c)

y = x⁴/4

d)

y = ¼(x² - 5)²

41.

Which of the following equations solves dy/dt = ky?

a)

y = y₀e^kt

b)

y = mx + b

c)

y = a(x - h)² + k

d)

y = Asin[B(x - C)] + D

42.

What movie is this photo from?

a)

Oliver & Company

b)

Lion King

c)

Lady & the Tramp

d)

Jungle Book

43.

What movie is this picture from?

a)

Little Mermaid

b)

Lady & the Tramp

c)

Bugs Life

d)

Frozen

44.

2. Solve the differential equation

𝑑𝑦/𝑑𝑡 =3𝑡²/𝑦 with initial condition

𝑦(2) = 0.

a)

$y=\ln\left|15t\right|$

b)

$y=16t^3$

c)

$y\ =\ \sqrt{2t^3-16}$

d)

$y=2t^3-16$

45.

A PDE consists of

a)

only one independent variable

b)

more than one independent variable

c)

more than one dependent varible

46.

The highest derivative in the given pde is called

a)

order

b)

degree

47.

The degree of the following pde $\left(\text{}\frac{\partial z}{\partial x}\ \right)^2+2\left(\frac{\partial^2z}{\partial x\partial y}\right)+\left(\frac{\partial z}{\partial y}\right)^2=0$ is

a)

2

b)

0

c)

1

48.

The order of the following pde $\left(z_{xx}\right)^2+\left(z_{xyy}\right)+\left(z_{yy}\right)=\sin z$ is

a)

1

b)

2

c)

3

49.

The expansion for s

a)

$\frac{\partial^2z}{\partial y^2}$

b)

$\frac{\partial^2z}{\partial x^2}$

c)

$\frac{\partial^2z}{\partial x\partial y^{ }}$

50.

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

a)

>1

b)

=1

c)

<1

51.

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

a)

<n

b)

>n

c)

=n

d)

1

52.

The complete integral is defined as

a)

The number of arbitrary constants < The number of independent variables

b)

The number of arbitrary constants = The number of independent variables

c)

The number of arbitrary constants > The number of independent variables

53.

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

a)

Particular solution

b)

General solution

c)

Singular solution

54.

which of the following has the trial solution z=ax+by+c

a)

$z=px+qy+p^2$

b)

$p+q=pq$

c)

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

55.

Which of the following is clairaut's form

a)

$z=px+qy+\sqrt{pq}$

b)

$p+q=1$

c)

$p+q=x+y$

56.

The complete integral of $F_1\left(x,p\right)=F_2\left(y,q\right)$ is

a)

$z=\int_{ }^{ }qdx+\int_{ }^{ }pdy$

b)

$z=\int_{ }^{ }pdx-\int_{ }^{ }qdy$

c)

$z=\int_{ }^{ }pdx+\int_{ }^{ }qdy$

57.

The subsidiary equation for Lagrange's linear equation

a)

$\frac{dx}{x}=\frac{dy}{y}=\frac{dz}{z}$

b)

$\frac{dx}{p}=\frac{dy}{q}=\frac{dz}{r}$

c)

$\frac{dx}{P}=\frac{dy}{Q}=\frac{dz}{R}$

58.

The multipliers for $x\left(y-z\right)p+y\left(z-x\right)q=z\left(x-y\right)$ is

a)

$1,1,1\ and\ \frac{1}{x},\frac{1}{y},\frac{1}{z}$

b)

$1,1,1\ and\ x,y,z$

c)

$0,0,0\ and\ 1,1,1$

59.

$The\ CF\ of\ \left(D^2-DD'-2D'^2\right)z=0$

a)

$f_1\left(y-2x\right)+f_2\left(y-x\right)$

b)

$f_1\left(y+2x\right)+f_2\left(y+x\right)$

c)

$f_1\left(y+2x\right)+f_2\left(y-x\right)$

60.

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

a)

z = p/q

b)

z = p - q

c)

z = p + q

d)

z = pq

61.

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

a)

First order PDE

b)

Second order PDE

c)

third order PDE

d)

Second order linear PDE

62.

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

a)

First order PDE

b)

Second and higher order PDE

c)

Second and higher order PDE

d)

only Second order linear PDE

63.

The Clairaut’s equation is of the form____

a)

y=px+f(p)

b)

y=px+f(q)

c)

x=qy+f(p)

d)

x=py+f(q)

64.

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

a)

p=a

b)

z=pq

c)

q=a

d)

p=aq

65.

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

a)

z = ax + by + pq

b)

z = ax + by + ab

c)

z = bx + ay + ab

d)

z = px + qy + ab

66.

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

a)

px = qy

b)

py+qx=0

c)

px+qy=0

d)

py = qx

67.

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

a)

px = qy

b)

px+qy=0

c)

py = qx

d)

py+qx=0

68.

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

a)

complete integral

b)

particular integral

c)

singular integral

d)

general integral

69.

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

a)

complete integral

b)

particular integral

c)

singular integral

d)

general integral

70.

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

a)

semi linear

b)

collinear

c)

Quasi linear

d)

non-linear

71.

A partial differential equation requires

a)

exactly one independent variable

b)

two or more independent variables

c)

more than one dependent variable

d)

equal number of dependent and independent variables

72.

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

a)

z = px + qy

b)

z = px - qy

c)

z = qx + py

d)

z = qx - py

73.

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

a)

p+q=0

b)

px = qy

c)

qx = py

d)

p-q=0

74.

$The\ solution\ of\ \frac{\partial^2z}{\partial y^2}=0$

a)

z = yf(x)+g(x)

b)

z = yf(y)+g(x)

c)

z = xf(x)+g(x)

d)

z = xf(y)+g(x)

75.

The solution z = ax + g(a)y +c is getting from

a)

f(x,p,q)=0

b)

f(p,q)=0

c)

f(x,p)=g(y,q)

d)

f(z,p,q)=0

76.

If the PDE is of the type f(p,q)=0, then we will take____

a)

f(y,q)=a

b)

f(x,p)=a

c)

p=q

d)

p=a

77.

If the PDE is of the type f(z,p,q)=0, then we will take____

a)

f(x,p)=a

b)

p=aq

c)

p=a

d)

p=q

78.

If the PDE is of the type f(x,p)=g(y,q), then we will take____

a)

f(x,p)=a=g(y,q)

b)

p=q

c)

p=aq

d)

p=a

79.

The solution of simultaneous linear differential equations is of the form___

a)

$\Phi\left(u,v\right)=0$

b)

$\Phi\left(u\right)=0$

c)

$\Phi\left(v\right)=0$

d)

$\Phi\left(x,y\right)=0$

80.

Find the complementary function of $\left(D^3-3D^2D'+2DD^{'2}\right)Z=0$

a)

$Z=f_1\left(y\right)+f_2\left(y+x\right)+f_3\left(y+2x\right)$

b)

$Z=f_1\left(y-x\right)+f_2\left(y+x\right)+f_3\left(y+2x\right)$

c)

$Z=f_1\left(y\right)+xf_2\left(y+x\right)+f_3\left(y+2x\right)$

d)

$Z=f_1\left(y\right)+f_2\left(y-x\right)+f_3\left(y-2x\right)$

81.

Find the roots of $m^3-m^2+m-1=0$

a)

1,i,-i

b)

1,-1,i

c)

i,-i,0

d)

1,-1,0

82.

Find the complementary function of $\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial x\partial y}-2\frac{\partial^2z}{\partial y^2}=0$

a)

$Z=f_1\left(y+x\right)+f_2\left(y-2x\right)$

b)

$Z=f_1\left(y-x\right)+f_2\left(y+2x\right)$

c)

$Z=f_1\left(y\right)+f_2\left(y+2x\right)$

d)

$Z=f_1\left(y+x\right)+xf_2\left(y-2x\right)$

83.

Find the complementary function of $\left(D^3-7DD'^2-6D'^3\right)z=0$

a)

$Z=f_1\left(y-x\right)+f_2\left(y-2x\right)+f_3\left(y+3x\right)$

b)

$Z=f_1\left(y+x\right)+f_2\left(y+2x\right)+f_3\left(y-3x\right)$

c)

$Z=f_1\left(y+x\right)+f_2\left(y-2x\right)+xf_3\left(y+3x\right)$

d)

$Z=f_1\left(y-x\right)+f_2\left(y\right)+f_3\left(y+3x\right)$

84.

Find the complementary function of $\frac{\partial^3z}{\partial x^3}-3\frac{\partial^3z}{\partial x^2\partial y}+4\frac{\partial^3z}{\partial y^3}=0$

a)

$Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+xf_3\left(y+2x\right)$

b)

$Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+f_3\left(y-2x\right)$

c)

$Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+f_3\left(y+2x\right)$

d)

$Z=f_1\left(y+x\right)+f_2\left(y+2x\right)+xf_3\left(y+2x\right)$

85.

Solve $\left(D^3-DD'^2-D'^3\right)z=0$

a)

$Z=f_1\left(y+x\right)+f_2\left(y-x\right)+xf_3\left(y-x\right)$

b)

$Z=f_1\left(y-2x\right)+xf_2\left(y-2x\right)+f_3\left(y+x\right)$

c)

$Z=f_1\left(y\right)+xf_2\left(y\right)+f_3\left(y-x\right)$

d)

$Z=f_1\left(y+x\right)+f_2\left(y-x\right)+f_3\left(y-x\right)$

86.

Solve $\frac{\partial^3z}{\partial x^3}-2\frac{\partial^3z}{\partial x^2\partial y}=0$

a)

$Z=f_1\left(y\right)+xf_2\left(y\right)+f_3\left(y+2x\right)$

b)

$Z=f_1\left(y\right)+f_2\left(y+x\right)+f_3\left(y+2x\right)$

c)

$Z=f_1\left(y\right)+f_2\left(y-x\right)+f_3\left(y+2x\right)$

d)

$Z=f_1\left(x\right)+yf_2\left(x\right)+f_3\left(y+2x\right)$

87.

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

a)

2,2

b)

-2,-2

c)

2,-2

d)

0,2

88.

Solve $\left(D^2+2DD'+D'^2\right)z=0$

a)

$Z=f_1\left(y-x\right)+xf_2\left(y-x\right)$

b)

$Z=f_1\left(y+x\right)+xf_2\left(y-x\right)$

c)

$Z=f_1\left(y+x\right)+xf_2\left(y+x\right)$

d)

$Z=f_1\left(y-x\right)+f_2\left(y-x\right)$

89.

Solve $\left(D^2-3DD'+2D'^2\right)z=0$

a)

$Z=f_1\left(y+x\right)+f_2\left(y+2x\right)$

b)

$Z=f_1\left(y\right)+f_2\left(y+x\right)$

c)

$Z=f_2\left(y-x\right)+f_3\left(y-2x\right)$

d)

$Z=xf_1\left(y\right)+xf_2\left(y+x\right)$

90.

The subsidiary equation for Lagrange's linear equation

a)

$\frac{dx}{x}=\frac{dy}{y}=\frac{dz}{z}$

b)

$\frac{dx}{p}=\frac{dy}{q}=\frac{dz}{r}$

c)

$\frac{dx}{P}=\frac{dy}{Q}=\frac{dz}{R}$

91.

The multipliers for $x\left(y-z\right)p+y\left(z-x\right)q=z\left(x-y\right)$ is

a)

$1,1,1\ and\ \frac{1}{x},\frac{1}{y},\frac{1}{z}$

b)

$1,1,1\ and\ x,y,z$

c)

$0,0,0\ and\ 1,1,1$

92.

Which of the following is clairaut's form

a)

$z=px+qy+\sqrt{pq}$

b)

$p+q=1$

c)

$p+q=x+y$

93.

The degree of the following pde $\left(\text{}\frac{\partial z}{\partial x}\ \right)^2+2\left(\frac{\partial^2z}{\partial x\partial y}\right)+\left(\frac{\partial z}{\partial y}\right)^2=0$ is

a)

2

b)

0

c)

1

94.

The solution of simultaneous linear differential equations is of the form___

a)

$\Phi\left(u,v\right)=0$

b)

$\Phi\left(u\right)=0$

c)

$\Phi\left(v\right)=0$

d)

$\Phi\left(x,y\right)=0$

95.

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

a)

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

b)

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

c)

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

d)

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

96.

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

a)

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

b)

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

c)

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

d)

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

97.

The solution of non-linear partial differential equation

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

is

a)

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

b)

$z=ax+ay+b$

c)

$z=ax+2ay+c$

d)

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

98.

The particular integral of the equation $\left(D-1\right)y=e^{3x}$ is (K1, CO1)

a)

$\frac{e^{3x}}{2}$

b)

$\frac{e^{-3x}}{2}$

c)

$\frac{e^{3x}}{4}$

d)

$\frac{e^x}{2}$

99.

The solution of non-linear partial differential equation

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

is

a)

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

b)

$z=ax+ay+b$

c)

$z=ax+2ay+c$

d)

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

100.

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

a)

$2,3,4$

b)

$3,4,2$

c)

$4,2,3$

d)

$1,1,1$

101.

What are the Lagrange’s multipliers while solving the PDE

p\left(y-z\right)-q\left(2x+y\right)=2x+z

a)

$x,y,z$

b)

$2x,y,z$

c)

$x,-2y,z$

d)

$2x,-z,-y$

102.

$.$ Find the P.I of $\left(D^3+D^2D'+DD'^2+D'^3\right)Z=0$

a)

$1$

b)

$0$

c)

$e^x$

d)

$none$

103.

$.$ Find the P.I of $\left(D^2-5DD'+6D'^2\right)z=e^{\left(x+y\right)}$

a)

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

b)

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

c)

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

d)

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

104.

$\ .$ Find the P.I of $\left(D^3-3D^2D'+4D'^2\right)Z=e^{\left(x+2y\right)}$

a)

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

b)

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

c)

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

d)

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

105.

.

Find the P.I of

\left(D^2-2DD'+D'^2\right)Z=\cos⁡(x-3y).

a)

$-\frac{1}{16}\sin⁡(x-3y)$

b)

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

c)

$\frac{1}{16}\cos⁡(x-3y)$

d)

$\frac{1}{16}\sin⁡(x-3y)$

106.

.

Find the P.I of

\left(D^2-4DD'+4D'^2\right)Z=e^{\left(2x+y\right)}

a)

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

b)

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

c)

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

d)

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

107.

.

Find the P.I of

\left(D^2+3DD'-4D'^2\right)Z=\sin⁡y.

a)

$\frac{1}{4}\cos⁡y$

b)

$\frac{1}{4}\cos⁡x$

c)

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

d)

$\frac{1}{4}\sin⁡y$

108.

.

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

a)

1

b)

2

c)

3

d)

4

109.

a)

b)

c)

d)

110.

The highest derivative in the given pde is called

a)

order

b)

degree

111.

The degree of the following pde $\left(\text{}\frac{\partial z}{\partial x}\ \right)^2+2\left(\frac{\partial^2z}{\partial x\partial y}\right)+\left(\frac{\partial z}{\partial y}\right)^2=0$ is

a)

2

b)

0

c)

1

112.

Which of the following is clairaut's form

a)

$z=px+qy+\sqrt{pq}$

b)

$p+q=1$

c)

$p+q=x+y$

113.

The complete integral of $F_1\left(x,p\right)=F_2\left(y,q\right)$ is

a)

$z=\int_{ }^{ }qdx+\int_{ }^{ }pdy$

b)

$z=\int_{ }^{ }pdx-\int_{ }^{ }qdy$

c)

$z=\int_{ }^{ }pdx+\int_{ }^{ }qdy$

114.

The subsidiary equation for Lagrange's linear equation

a)

$\frac{dx}{x}=\frac{dy}{y}=\frac{dz}{z}$

b)

$\frac{dx}{p}=\frac{dy}{q}=\frac{dz}{r}$

c)

$\frac{dx}{P}=\frac{dy}{Q}=\frac{dz}{R}$

115.

The multipliers for $x\left(y-z\right)p+y\left(z-x\right)q=z\left(x-y\right)$ is

a)

$1,1,1\ and\ \frac{1}{x},\frac{1}{y},\frac{1}{z}$

b)

$1,1,1\ and\ x,y,z$

c)

$0,0,0\ and\ 1,1,1$

116.

$The\ CF\ of\ \left(D^2-DD'-2D'^2\right)z=0$

a)

$f_1\left(y-2x\right)+f_2\left(y-x\right)$

b)

$f_1\left(y+2x\right)+f_2\left(y+x\right)$

c)

$f_1\left(y+2x\right)+f_2\left(y-x\right)$

117.

The partial differential equation

is classified as

a)

elliptic

b)

parabolic

c)

hyperbolic

d)

none of the above

118.

Solve $\left(D^3-3D^2D'+2DD^{'2}\right)Z=0$

a)

$Z=f_1\left(y\right)+f_2\left(y+x\right)+f_3\left(y+2x\right)$

b)

$Z=f_1\left(y-x\right)+f_2\left(y+x\right)+f_3\left(y+2x\right)$

c)

$Z=f_1\left(y\right)+xf_2\left(y+x\right)+f_3\left(y+2x\right)$

d)

$Z=f_1\left(y\right)+f_2\left(y-x\right)+f_3\left(y-2x\right)$

119.

$\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial x\partial y}-2\frac{\partial^2z}{\partial y^2}=0$

a)

$Z=f_1\left(y+x\right)+f_2\left(y-2x\right)$

b)

$Z=f_1\left(y-x\right)+f_2\left(y+2x\right)$

c)

$Z=f_1\left(y\right)+f_2\left(y+2x\right)$

d)

$Z=f_1\left(y+x\right)+xf_2\left(y-2x\right)$

120.

$\frac{\partial^3z}{\partial x^3}-3\frac{\partial^3z}{\partial x^2\partial y}+4\frac{\partial^3z}{\partial y^3}=0$

a)

$Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+xf_3\left(y+2x\right)$

b)

$Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+f_3\left(y-2x\right)$

c)

$Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+f_3\left(y+2x\right)$

d)

$Z=f_1\left(y+x\right)+f_2\left(y+2x\right)+xf_3\left(y+2x\right)$

121.

Solve $\left(D^3-DD'^2-D'^3\right)z=0$

a)

$Z=f_1\left(y+x\right)+f_2\left(y-x\right)+xf_3\left(y-x\right)$

b)

$Z=f_1\left(y-2x\right)+xf_2\left(y-2x\right)+f_3\left(y+x\right)$

c)

$Z=f_1\left(y\right)+xf_2\left(y\right)+f_3\left(y-x\right)$

d)

$Z=f_1\left(y+x\right)+f_2\left(y-x\right)+f_3\left(y-x\right)$

122.

Solve $\frac{\partial^3z}{\partial x^3}-2\frac{\partial^3z}{\partial x^2\partial y}=0$

a)

$Z=f_1\left(y\right)+xf_2\left(y\right)+f_3\left(y+2x\right)$

b)

$Z=f_1\left(y\right)+f_2\left(y+x\right)+f_3\left(y+2x\right)$

c)

$Z=f_1\left(y\right)+f_2\left(y-x\right)+f_3\left(y+2x\right)$

d)

$Z=f_1\left(x\right)+yf_2\left(x\right)+f_3\left(y+2x\right)$

123.

Solve $\left(D^2-3DD'+2D'^2\right)z=0$

a)

$Z=f_1\left(y+x\right)+f_2\left(y+2x\right)$

b)

$Z=f_1\left(y\right)+f_2\left(y+x\right)$

c)

$Z=f_2\left(y-x\right)+f_3\left(y-2x\right)$

d)

$Z=xf_1\left(y\right)+xf_2\left(y+x\right)$

124.

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

a)

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

b)

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

c)

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

d)

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

125.

Which of the following equations represents Clairaut’s partial differential equation?

a)

z=px+f(p,q)

b)

z=f(p,q)

c)

z=p+q+f(p,q)

d)

z=px+qy+f(p,q)

126.

The solution of non-linear partial differential equation

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

is

a)

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

b)

$z=ax+ay+b$

c)

$z=ax+2ay+c$

d)

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

127.

The solution of non-linear partial differential equation

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

is

a)

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

b)

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

c)

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

d)

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

128.

The solution of non-linear partial differential equation

q-p+x-y=0

is

a)

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

b)

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

c)

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

d)

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

129.

By usual notations what is the general form of Lagrange’s linear PDE.

a)

pdx+qdy=0

b)

Pp+Qq=R

c)

Pdx+Qdy=Rdz

d)

non of the above

130.

Find the complete integral of pq=1 .

a)

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

b)

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

c)

iii) z=ax+by+c

d)

(i)&(ii)

131.

The complete solution of partial differential equation involves

a)

complete function + indefinite integral

b)

Complete function + Particular Integral

c)

Complementary function + Particular Integral

d)

complementary function + definite integral

132.

Find the particular integral of (D² -2DD^' +D²)z=e^x+2y

a)

2e^x

b)

e^x+2y

c)

e^x-y

d)

3e^x+2y

133.

The particular integral of the equation (D²+DD^'-6D^'2)z = e^3x+y

a)

$\frac{e^{3x-y}}{6}$

b)

$\frac{e^{3x+y}}{5}$

c)

$\frac{e^{3x}}{6}$

d)

$\frac{e^{3x+y}}{6}$

134.

Find the Particular Integral of (D +4DD'-4D' )z = cos(x+y)

a)

$-\cos\left(x+y\right)$

b)

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

c)

$\cos\left(x-y\right)$

d)

$\frac{\cos\left(x+y\right)}{-2}$

135.

Which of the following is an example for first order linear partial differential equation?

a)

Lagrange’s Partial Differential Equation

b)

Clairaut’s Partial Differential Equation

136.

What is the nature of Lagrange’s linear partial differential equation?

a)

First-order, Second-degree

b)

Second-order, First-degree

c)

First-order, Third-degree

d)

First-order, First-degree

137.

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

a)

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

b)

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

c)

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

d)

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

138.

Which of the following equations represents Clairaut’s partial differential equation?

a)

z=px+f(p,q)

b)

z=f(p,q)

c)

z=p+q+f(p,q)

d)

z=px+qy+f(p,q)

139.

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

a)

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

b)

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

c)

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

d)

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

140.

The solution of non-linear partial differential equation

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

is

a)

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

b)

$z=ax+ay+b$

c)

$z=ax+2ay+c$

d)

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

141.

The solution of non-linear partial differential equation

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

is

a)

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

b)

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

c)

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

d)

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

142.

The solution of non-linear partial differential equation

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

is

a)

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

b)

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

c)

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

143.

The solution of non-linear partial differential equation

q-p+x-y=0

is

a)

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

b)

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

c)

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

d)

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

144.

Name

4 lines

145.

Roll number(GU)

4 lines

146.

Registered mobile number

4 lines

147.

Name

4 lines

148.

e-mail id

4 lines

149.

Contact No.

4 lines

150.

Organization

4 lines

151.

Position

4 lines

152.

How would you rate the overall webinar experience?
(undefined = 0, undefined = 4)

a)

0

b)

1

c)

2

d)

3

e)

4

153.

How would you rate the content of the webinar?
(undefined = 0, undefined = 4)

a)

0

b)

1

c)

2

d)

3

e)

4

Quiz on P. D. E 6th semester

Total questions: 153

Worksheet time: 3hrs 29mins

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