Fourier Analysis and Partial Differential Equations Multiple Choice Questions on “Solution of Second Order P.D.E.”.
1. What is the general form of second order non-linear partial differential equations (x and y being independent variables and z being a dependent variable)?
a) (F(x,y,z,frac{∂z}{∂x},frac{∂z}{∂y},frac{∂^2 z}{∂x^2},frac{∂^2 z}{∂y^2},frac{∂^2 z}{∂x∂y})=0)
b) (F(x,z,frac{∂z}{∂x},frac{∂z}{∂y},frac{∂^2 z}{∂x^2},frac{∂^2 z}{∂y^2})=0)
c) (F(y,z,frac{∂z}{∂x},frac{∂z}{∂y})=0)
d) F(x,y)=0
View Answer
Answer: a
Explanation: The most general second order partial differential equation in two independent variables x and y, and z as the dependent variable has the form,
(F(x,y,z,frac{∂z}{∂x},frac{∂z}{∂y},frac{∂^2 z}{∂x^2},frac{∂^2 z}{∂y^2},frac{∂^2 z}{∂x∂y})=0)
2. The solution of the general form of second order non-linear partial differential equation is obtained by Monge’s method.
a) False
b) True
View Answer
Answer: b
Explanation: Gaspard Monge (9 May 1746 – 28 July 1818) was a French mathematician, the inventor of descriptive geometry (the mathematical basis of technical drawing), and the father of differential geometry.
3. What is the reason behind the non-existence of any real function which satisfies the differential equation, (y’)2 + 1 = 0?
a) Because for any real function, the left-hand side of the equation will be less than, or equal to one and thus cannot be zero
b) Because for any real function, the left-hand side of the equation becomes zero
c) Because for any real function, the left-hand side of the equation will be greater than, or equal to one and thus cannot be zero
d) Because for any real function, the left-hand side of the equation becomes infinity
View Answer
Answer: c
Explanation: Given: (y’)2 + 1 = 0
Consider if y = 2x, then y’ = 2 and hence the left-hand side of the equation becomes 3 which is greater than 1. Therefore, the left-hand side of the equation will always be greater than, or equal to one and thus cannot be zero and hence the differential equation is not satisfied.
4. What is the order of the partial differential equation, (frac{∂^2 z}{∂x^2}-(frac{∂z}{∂y})^5+frac{∂^2 z}{∂x∂y}=0)?
a) Order-5
b) Order-1
c) Order-4
d) Order-2
View Answer
Answer: d
Explanation: The order of an equation is defined as the highest derivative present in the equation. Hence, in the given equation, (frac{∂^2 z}{∂x^2}-(frac{∂z}{∂y})^5+frac{∂^2 z}{∂x∂y}=0), the order is 2.
5. Which of the following is the condition for a second order partial differential equation to be hyperbolic?
a) b2-ac<0
b) b2-ac=0
c) b2-ac>0
d) b2-ac=<0
View Answer
Answer: c
Explanation: For a second order partial differential equation to be hyperbolic, the equation should satisfy the condition, b2-ac>0.
6. Which of the following represents the canonical form of a second order parabolic PDE?
a) (frac{∂^2 z}{∂η^2}+⋯=0 )
b) (frac{∂^2 z}{∂ζ∂η}+⋯=0)
c) (frac{∂^2 z}{∂α^2}+frac{∂^2 z}{∂β^2}…=0)
d) (frac{∂^2 z}{∂ζ^2}+⋯=0)
View Answer
Answer: a
Explanation: A second order linear partial differential equation can be reduced to so-called canonical form by an appropriate change of variables ξ = ξ(x, y), η = η(x, y).
7. The condition which a second order partial differential equation must satisfy to be elliptical is
b2-ac=0.
a) True
b) False
View Answer
Answer: b
Explanation: The condition for a second order partial differential equation to be elliptical is given by, b2-ac<0.
8. Which of the following statements is true?
a) Hyperbolic equations have three families of characteristic curves
b) Hyperbolic equations have one family of characteristic curves
c) Hyperbolic equations have no families of characteristic curves
d) Hyperbolic equations have two families of characteristic curves
View Answer
Answer: d
Explanation: The canonical variables ξ and η for a hyperbolic pde satisfy the equations,
(aζ_x+(b+sqrt{b^2-ac}) ζ_y=0 , and , aη_x+(b+sqrt{b^2-ac}) η_y=0)
The families of curves ξ = constant and η = constant are the characteristic curves. Hence, hyperbolic equations have two families of characteristic curves.
9. Which of the following represents the family of the characteristic curves for parabolic equations?
a) aζx+bζy=0
b) aζx+b=0
c) a+ζy=0
d) a(ζx+ζy)=0
View Answer
Answer: a
Explanation: Parabolic equations have only one family of characteristic curves. Instead of two equations like for hyperbolic equations, we have just the single equation, aζx+bζy=0 (or aηx+bηy=0).
10. The condition that a second order partial differential equation should satisfy to be parabolic is b2-ac=0.
a) True
b) False
View Answer
Answer: a
Explanation: If the second order partial differential equation satisfies the condition, b2-ac=0, then it is said to be parabolic in nature.
11. Elliptic equations have no characteristic curves.
a) True
b) False
View Answer
Answer: a
Explanation: For an elliptic equation, b2 −ac < 0 so equations obtained contain complex coefficients and
have no real solutions. Hence, elliptic equations have no characteristic curves.
12. Singular solution of a differential equation is one that cannot be obtained from the general solution gotten by the usual method of solving the differential equation.
a) True
b) False
View Answer
Answer: a
Explanation: A differential equation is said to have a singular solution if in all points in the domain of the equation the uniqueness of the solution is violated. Hence, this solution cannot be obtained from the general solution.
13. In the formation of differential equation by elimination of arbitrary constants, after differentiating the equation with respect to independent variable, the arbitrary constant gets eliminated.
a) False
b) True
View Answer
Answer: a
Explanation: In the formation of differential equation by elimination of arbitrary constants, the first step is to differentiate the equation with respect to the dependent variable. Sometimes, the arbitrary constant gets eliminated after differentiation.
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