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

Ordinary Partial Differential Equation

Partially Ordinary Differential Equation

more than one independent variable

more than one dependent varible

The complete integral is defined as

The number of arbitrary constants = The number of independent variables

The number of arbitrary constants < The number of independent variables

The number of arbitrary constants > The number of independent variables

Which of the following is clairaut's form

z = p x + q y + p q z=px+qy+\sqrt{pq} z = p x + q y + p q <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

p + q = x + y p+q=x+y p + q = x + y

The complete integral of F 1 ( x , p ) = F 2 ( y , q ) F_1\left(x,p\right)=F_2\left(y,q\right) F 1 ( x , p ) = F 2 ( y , q ) is

z = ∫ p d x + ∫ q d y z=\int_{ }^{ }pdx+\int_{ }^{ }qdy z = ∫ p d x + ∫ q d y

z = ∫ q d x + ∫ p d y z=\int_{ }^{ }qdx+\int_{ }^{ }pdy z = ∫ q d x + ∫ p d y

z = ∫ p d x − ∫ q d y z=\int_{ }^{ }pdx-\int_{ }^{ }qdy z = ∫ p d x − ∫ q d y

The subsidiary equation for Lagrange's linear equation

d x P = d y Q = d z R \frac{dx}{P}=\frac{dy}{Q}=\frac{dz}{R} P d x = Q d y = R d z

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

d x p = d y q = d z r \frac{dx}{p}=\frac{dy}{q}=\frac{dz}{r} p d x = q d y = r d z d x p = d y q = d z r \frac{dx}{p}=\frac{dy}{q}=\frac{dz}{r} p d x = q d y = r d z

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

1 , 1 , 1 a n d 1 x , 1 y , 1 z 1,1,1\ and\ \frac{1}{x},\frac{1}{y},\frac{1}{z} 1 , 1 , 1 a n d x 1 , y 1 , z 1

1 , 1 , 1 a n d x , y , z 1,1,1\ and\ x,y,z 1 , 1 , 1 a n d x , y , z

0 , 0 , 0 a n d 1 , 1 , 1 0,0,0\ and\ 1,1,1 0 , 0 , 0 a n d 1 , 1 , 1

A partial differential equation requires

two or more independent variables

exactly one independent variable

more than one dependent variable

equal number of dependent and independent variables

A partial differential equation requires

two or more independent variables

exactly one independent variable

more than one dependent variable

equal number of dependent and independent variables

The complete integral is defined as

The number of arbitrary constants = The number of independent variables

The number of arbitrary constants < The number of independent variables

The number of arbitrary constants > The number of independent variables

PDE First and Higher Order Quiz

Authored by Athithan S

Mathematics

University

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MULTIPLE CHOICE QUESTION

30 sec • 1 pt

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

Lagrange’s Partial Differential Equation

Clairaut’s Partial Differential Equation

Answer explanation

Lagrange’s Partial Differential Equation is an example of a first-order linear partial differential equation.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

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

First-order, Second-degree

Second-order, First-degree

First-order, Third-degree

First-order, First-degree

Answer explanation

Lagrange's linear partial differential equation is First-order, First-degree.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

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

Answer explanation

The correct choice is phi(x^2-y^2, y^2-z^2)=0 because it represents the general solution of the given linear partial differential equation.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

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

z=px+f(p,q)

z=f(p,q)

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

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

Answer explanation

The correct choice is z=px+qy+f(p,q) because it represents Clairaut’s partial differential equation.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

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

Answer explanation

The correct solution of Lagrange's partial differential equation xp+yq=z is f(x/y, y/z)=0.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Answer explanation

The correct choice is z=ax+(1-√a)^2y+c because it satisfies the given non-linear partial differential equation.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Answer explanation

The correct choice is z=(x-a)^3/3+a^2y+b because it satisfies the given non-linear partial differential equation.

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PDE First and Higher Order Quiz

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

Lagrange’s Partial Differential Equation is an example of a first-order linear partial differential equation.

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

Lagrange's linear partial differential equation is First-order, First-degree.

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

The correct choice is phi(x^2-y^2, y^2-z^2)=0 because it represents the general solution of the given linear partial differential equation.

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

The correct choice is z=px+qy+f(p,q) because it represents Clairaut’s partial differential equation.

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

The correct solution of Lagrange's partial differential equation xp+yq=z is f(x/y, y/z)=0.

The correct choice is z=ax+(1-√a)^2y+c because it satisfies the given non-linear partial differential equation.

The correct choice is z=(x-a)^3/3+a^2y+b because it satisfies the given non-linear partial differential equation.

The correct choice is \(\log\left(az-1\right)=x+a\ y+c\) because it represents the solution of the given non-linear partial differential equation.

The correct choice is z=(x+a)^2/2+(y+a)^2/2+c, which satisfies the given non-linear partial differential equation.

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

A Partial Differential Equation involves partial derivatives of dependent variables with respect to multiple independent variables.

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