This set of Partial Differential Equations Questions and Answers for Entrance exams focuses on “Method of Separation of Variables”.
1. By using the method of separation of variables, the determination of solution to P.D.E. reduces to determination of solution to O.D.E.
a) True
b) False
View Answer
Explanation: A differential equation is an equation involving an unknown function y of one or more independent variables x, t, …… and its derivatives. These are divided into two types, ordinary or partial differential equations.
An ordinary differential equation is a differential equation in which a dependent variable (say ‘y’) is a function of only one independent variable (say ‘x’).
A partial differential equation is one in which a dependent variable depends on one or more independent variables.
2. Separation of variables, in mathematics, is also known as Fourier method.
a) False
b) True
View Answer
Explanation: Separation of variables (or the Fourier method) is a method of solving ordinary and partial differential equations, in which we can rewrite an equation so that each of two variables occur on different sides of the equation.
3. Which of the following equations cannot be solved by using the method of separation of variables?
a) Laplace Equation
b) Helmholtz Equation
c) Alpha Equation
d) Biharmonic Equation
View Answer
Explanation: The method of separation of variables is used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as:
- Heat equation
- Wave equation
- Laplace equation
- Helmholtz equation
- Biharmonic equation
4. The matrix form of the separation of variables is the Kronecker sum.
a) True
b) False
View Answer
Explanation: If A is n × n, B is m × m and Ik denotes the k × k identity matrix then we can define what is sometimes called the Kronecker sum, ⊕, by,
A⊕B=A⊗IM+B⊗IN
5. For a partial differential equation, in a function φ (x, y) and two variables x, y, what is the form obtained after separation of variables is applied?
a) Φ (x, y) = X(x)+Y(y)
b) Φ (x, y) = X(x)-Y(y)
c) Φ (x, y) = X(x)Y(y)
d) Φ (x, y) = X(x)/Y(y)
View Answer
Explanation: The method of separation of variables relies upon the assumption that a function of the form,
Φ (x, y) = X(x)Y(y)
will be a solution to a linear homogeneous partial differential equation in x and y. This is called a product solution and provided the boundary conditions are also linear and homogeneous this will also satisfy the boundary conditions.
6. What is the solution of, \(\frac{∂^2 u}{∂x^2}=2xe^t,\) after applying method of separation of variables \((u(0,t)=t,\frac{∂u}{∂x} (0,t)= e^t)\)?
a) \(u=\frac{x^3}{3} e^t+xe^t \)
b) \(u=\frac{x^3}{3} e^t+xe^t+t\)
c) \(u=\frac{x^3}{3} e^t+e^t+t\)
d) \(u=\frac{x^2}{2} e^t+xe^t+t\)
View Answer
Explanation: Given: \(\frac{∂^2 u}{∂x^2}=2xe^t,\)……………………………………………………………………………… (1)
Integrating (1) with respect to x we get,
\(\frac{∂u}{∂x}=x^2 e^t+f(t),\) where f(t) = arbitrary function ………………………………. (2)
Integrating again with respect to x we get,
\(u=\frac{x^3}{3} e^t+xf(t)+g(t)\)………………………………………………………………………………………… (3)
Applying the given initial condition, \(\frac{∂u}{∂x}(0,t)= e^t\) in equation (2), we get, f(t)=et.
Applying the initial condition, u(0,t)=t in equation (3), we get, g(t)=t.
Therefore, the solution, \(u=\frac{x^3}{3} e^t+xe^t+t.\)
7. Which of the following is true with respect to formation of differential equation by elimination of arbitrary constants?
a) The given equation should be differentiated with respect to independent variable
b) Elimination of the arbitrary constant by replacing it using derivative
c) If ‘n’ arbitrary constant is present, the given equation should be differentiated ‘n’ number of times
d) To eliminate the arbitrary constants, the given equation must be integrated with respect to the dependent variable
View Answer
Explanation: Consider a general equation, f(x,y,c)=0 ……………………………………… (1)
To form a differential equation by elimination of arbitrary constant, the following steps need to be followed:
- Differentiate (1) with respect to x
- In case of ‘n’ arbitrary constants, the equation should be differentiated ‘n’ number of times
- Eliminate the arbitrary constant using (1) and the derivatives
8. 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
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.
9. u (x, t) = e − 2π*2t*sin πx is the solution of the two-dimensional Laplace equation.
a) True
b) False
View Answer
Explanation: The solution of the two-dimensional Laplace equation is,
u(x, y) = sin x*cosh y
10. The symbol used for partial derivatives, ∂, was first used in mathematics by Marquis de Condorcet.
a) True
b) False
View Answer
Explanation: Partial derivatives are indicated by the symbol ∂. This was first used in mathematics by Marquis de Condorcet who used it for partial differences.
11. Separation of variables was first used by L’Hospital in 1750.
a) False
b) True
View Answer
Explanation: Guillaume François Antoine, Marquis de L’Hospital (1661 – 2 February 1704), was a French mathematician. His name is firmly associated with L’Hospital’s rule for calculating limits involving indeterminate forms 0/0 and ∞/∞.
12. A particular solution for an equation is derived by eliminating arbitrary constants.
a) True
b) False
View Answer
Explanation: A particular solution for an equation is derived by substituting particular values to the arbitrary constants in the complete solution.
Sanfoundry Global Education & Learning Series – Fourier Analysis and Partial Differential Equations.
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