Quiz about differential equation and application -quiz 2

Question 1

with the usual notation the differential equation 2 r + 5 s + 2 t = 0 is same as .....................

Accepted Answer

$$2\frac{\partial^2z}{\partial x^2}+5\frac{\partial^2z}{\partial x\partial y}+2\frac{\partial^2z}{\partial y^2}=0$$

Answer

$$2\ \frac{\partial z}{\partial x}+5\frac{\partial z}{\partial y}+2\frac{\partial^2z}{\partial x\partial y}=0$$

Answer

$$2\left(\frac{\partial z^2}{\partial x^2} ight)+5\left(\frac{\partial x}{\partial y} ight)+ ext{5}\frac{\partial x^2}{\partial y^2}=\ 0$$

Answer

$$2\frac{ ext{d}^2x}{ ext{d}x^2}+5\frac{ ext{d}^2z}{ ext{d}y ext{d}x}+2\frac{ ext{d}^2z}{ ext{d}y^2}=0$$

Question 2

Charpit's method gives...................

Accepted Answer

solutions to first-order partial differential equations.

Answer

solutions to linear algebraic equations.

Answer

numerical approximations for integrals.

Answer

solutions to second-order differential equations.

Question 3

The orthogonal trajectories of the family of circles $$x^2+y^2=a^2$$ is

Accepted Answer

The family of straight lines passing through the origin

Answer

The family of circles passing through the origin

Answer

the family of straight lines not passing through the origin

Answer

none of the above

Question 4

The auxiliary equation of the partial differential equation 2p+3q = 1 is

Accepted Answer

$$\frac{ ext{d}x}{ ext{2}}=\frac{ ext{d}y}{ ext{3}}=\frac{ ext{d}z}{ ext{1}}$$

Answer

$$\frac{ ext{d}x}{ ext{2}}+\frac{ ext{d}y}{ ext{3}}=\frac{ ext{d}z}{ ext{4}}$$

Answer

$$\frac{ ext{d}x}{ ext{2}} ext{} ext{=}\frac{ ext{d}y}{ ext{1}}=\frac{ ext{d}z}{ ext{0}}$$

Answer

none of the above

Question 5

The partial differential equation by eliminating the arbitrary constants a and b from

z = axy + b is........................

Accepted Answer

$$px-qy\ =0$$

Answer

$$px+qy=0$$

Answer

$$py+qx=0$$

Answer

$$py-qx\ =0$$

Question 6

A solution obtained by giving particular values to the arbitrary constants in a complete integrals is called............................

Accepted Answer

particular integral

Answer

complete integral

Answer

singular integral

Answer

general integral

Question 7

The partial differential equation by eliminating the arbitrary constants a and b from $$z=ax+by+ab$$ is

Accepted Answer

$$px+qy+pq$$

Answer

$$p+q+pq$$

Answer

$$px+qy-pq$$

Answer

$$px+qy$$

Question 8

what is the complementary function of (D²+1) y = x

Accepted Answer

a cos x + b sin x

Answer

a cos 2x + b sin 2x

Answer

a $$e^x$$ +b $$e^{-x}$$

Answer

axe^3x+be^3x

Question 9

The general solution of p + q = 1 is ...............

Accepted Answer

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

Answer

$$\phi\left(x+z\ ,\ y+z\right)=0$$

Answer

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

Answer

none

Question 10

Say True or False:

A family of curves that intersect each curve of the given family orthogonally is said to be the family of orthogonal trajectories.

Accepted Answer

true

Answer

false

Question 11

The PDE obtained by eliminating a and b from $$az\ +b=a^2x+y$$ is............

Accepted Answer

pq = 1

Answer

p + q = - 1

Answer

z = p²x + q

Answer

none

Question 12

A PDE consists of

Accepted Answer

more than one independent variable

Answer

only one independent variable

Answer

more than one dependent varible

Question 13

The equation for Lagrange's linear equation

Accepted Answer

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

Answer

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

Answer

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

Question 14

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 15

The expansion for s

Accepted Answer

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

Answer

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

Answer

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

Question 16

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

Accepted Answer

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

Answer

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

Answer

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

Question 17

(D²+4D+4)y=0 has no complementary function.

Accepted Answer

False

Answer

True

Question 18

Say True or False:

The general solution of the lagrange's linear equation Pp+Qq=R is $$\phi\left(u-1,v+1\right)=0$$

Accepted Answer

False

Answer

True

Question 19

Find the auxiliary equation of zp + x =0

Accepted Answer

$$\frac{ ext{d}x}{ ext{z}}=\frac{ ext{d}y}{ ext{0}}=\frac{ ext{d}z}{ ext{-x}}$$

Answer

$$\frac{ ext{d}x}{ ext{x}}=\frac{ ext{d}y}{ ext{y}}=\frac{ ext{d}z}{ ext{z}}$$

Answer

$$\frac{ ext{d}z}{ ext{y}}=\frac{ ext{d}y}{ ext{0}}=\frac{ ext{d}x}{ ext{-z}}$$

Answer

none

Question 20

True or False:

The lagrangian multiplier for $$\left(y^2+z^2\right)p-xyq\ +xz\ =0$$ is x, y, z

Accepted Answer

True

Answer

False

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