Characteristics and PDE Classification - Partial Differential Equations - Mathematics Previous Year Questions

1

2008 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2008

The solution of \(xu_x + yu_y = 0\) is of the form

A

\(f(y/x)\)

B

\(f(x + y)\)

C

\(f(x - y)\)

D

\(f(xy)\)

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2

2008 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2008

If the partial differential equation \((x-1)^2 u_{xx} - (y-2)^2 u_{yy} + 2xu_x + 2yu_y + 2xyu = 0\) is parabolic in \(S \subseteq \mathbb{R}^2\) but not in \(\mathbb{R}^2 \setminus S\), then \(S\) is

A

\(\{(x, y) \in \mathbb{R}^2 : x = 1 \text{ or } y = 2\}\)

B

\(\{(x, y) \in \mathbb{R}^2 : x = 1 \text{ and } y = 2\}\)

C

\(\{(x, y) \in \mathbb{R}^2 : x = 1\}\)

D

\(\{(x, y) \in \mathbb{R}^2 : y = 2\}\)

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3

2008 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2008

The characteristic curve of \(2y u_x + (2x + y^2) u_y = 0\) passing through \((0, 0)\) is

A

\(y^2 = 2(e^x + x - 1)\)

B

\(y^2 = 2(e^x - x + 1)\)

C

\(y^2 = 2(e^x - x - 1)\)

D

\(y^2 = 2(e^x + x + 1)\)

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4

2008 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2008

The initial value problem \(u_x + u_y = 1, u(x, s) = \sin x, 0 \leq s \leq 1\), has

A

two solutions

B

a unique solution

C

no solution

D

infinitely many solutions

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5

2009 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2009

The integral surface satisfying the equation \( y \frac{\partial z}{\partial x} + x \frac{\partial z}{\partial y} = x^2 + y^2 \) and passing through the curve \( x = 1-t, \ y = 1+t, \ z = 1+t^2 \) is

A

\(z = xy + \frac{1}{2} (x^2 - y^2)^2 \)

B

\(z = xy + \frac{1}{4} (x^2 - y^2)^2 \)

C

\(z = xy + \frac{1}{8} (x^2 - y^2)^2 \)

D

\(z = xy + \frac{1}{16} (x^2 - y^2)^2 \)

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6

2010 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2010

The general solution of the partial differential equation \( \frac{\partial^2 z}{\partial x \partial y} = x + y \) is of the form

A

\(\frac{1}{2}xy(x + y) + F(x) + G(y) \)

B

\(\frac{1}{2}xy(x - y) + F(x) + G(y) \)

C

\(\frac{1}{2}xy(x - y) + F(x)G(y) \)

D

\(\frac{1}{2}xy(x + y) + F(x)G(y) \)

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7

2011 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2011

The partial differential equation \[x^2 \frac{\partial^2 z}{\partial x^2} - (y^2 - 1)x \frac{\partial^2 z}{\partial x \partial y} + y(y-1)^2 \frac{\partial^2 z}{\partial y^2} + x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 0\] is hyperbolic in a region in the \(XY\)-plane if

A

\(x \neq 0\) and \(y = 1\)

B

\(x = 0\) and \(y \neq 1\)

C

\(x \neq 0\) and \(y \neq 1\)

D

\(x = 0\) and \(y = 1\)

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8

2011 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2011

The integral surface for the Cauchy problem \( \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 1, \) which passes through the circle \( z = 0, x^2 + y^2 = 1 \) is

A

x^2 + y^2 + 2z^2 + 2zx - 2yz - 1 = 0

B

x^2 + y^2 + 2z^2 + 2zx + 2yz - 1 = 0

C

x^2 + y^2 + 2z^2 - 2zx - 2yz - 1 = 0

D

x^2 + y^2 + 2z^2 + 2zx + 2yz + 1 = 0

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9

2012 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2012

The integral surface satisfying the partial differential equation \(\frac{\partial z}{\partial x}+z^2\frac{\partial z}{\partial y}=0\) and passing through the straight line \(x=1,\ y=z\) is

A

\((x-1)z+z^2=y^2\)

B

\(x^2+y^2-z^2=1\)

C

\((y-z)x+z^2=1\)

D

\((x-1)z^2+z=y\)

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10

2014 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2014

The integral surface of the first order partial differential equation \(2y(z-3)\frac{\partial z}{\partial x} + (2x-z)\frac{\partial z}{\partial y} = y(2x-3)\) passing through the curve \(x^2 + y^2 = 2x, z = 0\) is

A

\(x^2 + y^2 - z^2 - 2x + 4z = 0\)

B

\(x^2 + y^2 - z^2 - 2x + 8z = 0\)

C

\(x^2 + y^2 + z^2 - 2x + 16z = 0\)

D

\(x^2 + y^2 + z^2 - 2x + 8z = 0\)

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11

2016 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2016

Let a, b, c, d ∈ ℝ such that c² + d² ≠ 0. Then, the Cauchy problem
a u_x + b u_y = e^x+y, x, y ∈ ℝ,
u(x,y) = 0 on c x + d y = 0
has a unique solution if

A

a c + b d ≠ 0

B

a d − b c ≠ 0

C

a c − b d ≠ 0

D

a d + b c ≠ 0

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12

2017 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2017

The partial differential equation \(x \frac{\partial^2 u}{\partial x^2} + (x - y) \frac{\partial^2 u}{\partial x \partial y} - y \frac{\partial^2 u}{\partial y^2} + \frac{1}{4} \left( \frac{\partial u}{\partial y} - \frac{\partial u}{\partial x} \right) = 0\) is

A

hyperbolic along the line \(x + y = 0\)

B

elliptic along the line \(x - y = 0\)

C

elliptic along the line \(x + y = 0\)

D

parabolic along the line \(x + y = 0\)

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13

2017 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2017

Let \( u(x,y) \) be the solution of \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 4u \) satisfying the condition \( u(x,y) = 1 \) on the circle \( x^2 + y^2 = 1 \). Then \( u(2,2) \) equals ________.

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14

2018 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2018

If the characteristic curves of the partial differential equation \(xu_{xx} + 2x^2u_{xy} = u_x - 1\) are \(\mu(x,y) = c_1\) and \(\nu(x,y) = c_2\), where \(c_1\) and \(c_2\) are constants, then

A

\(\mu(x,y) = x^2 - y, \ \nu(x,y) = y\)

B

\(\mu(x,y) = x^2 + y, \ \nu(x,y) = y\)

C

\(\mu(x,y) = x^2 + y, \ \nu(x,y) = x^2\)

D

\(\mu(x,y) = x^2 - y, \ \nu(x,y) = x^2\)

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15

2019 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2019

The partial differential equation
\( (x^2 + y^2 -1) \frac{\partial^2 u}{\partial x^2} + 2 \frac{\partial^2 u}{\partial x \partial y} + (x^2 + y^2 -1) \frac{\partial^2 u}{\partial y^2} = 0 \)
is

A

parabolic in the region \(x^2 + y^2 > 2 \)

B

hyperbolic in the region \(x^2 + y^2 > 2 \)

C

elliptic in the region \(0 < x^2 + y^2 < 2 \)

D

hyperbolic in the region \(0 < x^2 + y^2 < 2 \)

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16

2019 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2019

Let the general integral of the partial differential equation \((2xy - 1) \frac{\partial z}{\partial x} + (z - 2x^2) \frac{\partial z}{\partial y} = 2(x - yz)\) be given by \(F(u, v) = 0\), where \(F: \mathbb{R}^2 \to \mathbb{R}\) is a continuously differentiable function. ( \(\mathbb{R}\) is the set of all real numbers and \(\mathbb{R}^2 = \{ (x, y) : x, y \in \mathbb{R} \}\) ) Then which one of the following is TRUE?

A

\(u = x^2 + y^2 + z, v = xz + y\)

B

\(u = x^2 + y^2 - z, v = xz - y\)

C

\(u = x^2 - y^2 + z, v = yz + x\)

D

\(u = x^2 + y^2 - z, v = yz - x\)

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17

2020 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2020

Let \(u(x,y)\) be the solution of the initial value problem
\[\frac{\partial u}{\partial x} + (\sqrt{u}) \frac{\partial u}{\partial y} = 0, \quad u(x,0) = 1 + x^2.\]
Then the value of \(u(0,1)\) is __________ (rounded off to three decimal places)

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18

2021 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2021

Consider the second-order partial differential equation (PDE) \(\frac{\partial^2 u}{\partial x^2} + 4 \frac{\partial^2 u}{\partial x \partial y} + (x^2 + 4y^2) \frac{\partial^2 u}{\partial y^2} = \sin(x + y)\).
Consider the following statements:
P: The PDE is parabolic on the ellipse \(\frac{x^2}{4} + y^2 = 1\).
Q: The PDE is hyperbolic inside the ellipse \(\frac{x^2}{4} + y^2 = 1\).
Then

A

both P and Q are TRUE

B

P is TRUE and Q is FALSE

C

P is FALSE and Q is TRUE

D

both P and Q are FALSE

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19

2021 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2021

If \(u(x, y)\) is the solution of the Cauchy problem \(x \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 1,\ u(x, 0) = -x^2,\ x > 0\), then \(u(2, 1)\) is equal to

A

\(1 - 2e^{-2}\)

B

\(1 + 4e^{-2}\)

C

\(1 - 4e^{-2}\)

D

\(1 + 2e^{-2}\)

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20

2022 · Mathematics · Partial Differential Equations · Characteristics and PDE Classification

Mathematics (MA) 2022

The partial differential equation \[\frac{\partial^2 u}{\partial x^2} + 16 \frac{\partial^2 u}{\partial x \partial y} + 4 \frac{\partial^2 u}{\partial y^2} = 0\] is transformed to \[A \frac{\partial^2 u}{\partial \xi^2} + B \frac{\partial^2 u}{\partial \xi \partial \eta} + C \frac{\partial^2 u}{\partial \eta^2} = 0,\] using \(\xi = y - 2x\) and \(\eta = 7y - 2x\). Then, the value of \(\frac{1}{12^3}(B^2 - 4AC)\) is __________.

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