This set of Partial Differential Equations Assessment Questions and Answers focuses on “Homogeneous Linear PDE with Constant Coefficient”.

1. Homogeneous Equations are those in which the dependent variable (and its derivatives) appear in terms with degree exactly one.

a) True

b) False

View Answer

Explanation: Linear partial differential equations can further be classified as:

- Homogeneous for which the dependent variable (and its derivatives) appear in terms with degree

exactly one, and - Non-homogeneous which may contain terms which only depend on the independent variable

2. Which of the following is false regarding quasilinear equations?

a) All the terms with highest order derivatives of dependent variables occur linearly

b) The coefficients of terms with highest order derivatives of dependent variables are functions of only lower order derivatives of the dependent variables

c) Lower order derivatives can occur in any manner

d) All the terms with lower order derivatives of dependent variables occur linearly

View Answer

Explanation: A PDE is called as a quasi-linear if at the minimum one coefficient of the partial derivatives is a function of the dependent variable. For example, \(\frac{∂^2 u}{∂x^2}-u \frac{∂^2 u}{∂y^2}=0. \)

3. Which of the following is false regarding Bessel polynomials?

a) Krall and Fink (1949) defined the Bessel polynomials

b) The polynomials satisfy the recurrence formula

c) The solutions of homogeneous equations are closely related to Bessel polynomials

d) Bessel polynomials are not an orthogonal sequence of polynomials

View Answer

Explanation: The Bessel polynomials are an orthogonal sequence of polynomials. The most accepted definition for these series was put forth by Krall and Frink (1948) as,

\(y_n (x)=∑_{k=0}^n\frac{(n+k)!}{(n-k)!k!} (\frac{x}{2})^k.\)

4. Which of the following is not a homogeneous equation?

a) \(\frac{∂^2 u}{∂t^2}-c^2 \frac{∂^2 u}{∂x^2}=0\)

b) \(\frac{∂^2 u}{∂x^2}+\frac{∂^2 u}{∂y^2}=0\)

c) \(\frac{∂^2 u}{∂x^2}+(\frac{∂^2 u}{∂x∂y})^2+\frac{∂^2 u}{∂y^2}=x^2+y^2\)

d) \(\frac{∂u}{∂t} -T\frac{∂^2 u}{∂x^2}=0\)

View Answer

Explanation: As we know that homogeneous equations are those in which the dependent variable (and its derivatives) appear in terms with degree exactly one, hence the equation,

\(\frac{∂^2 u}{∂x^2}+(\frac{∂^2 u}{∂x∂y})^2+\frac{∂^2 u}{∂y^2}=x^2+y^2\) is not a homogenous equation (since its degree is 2).

5. What is the general solution of the DE with n linearly independent solutions u1(t), …., un(t) of a nth order linear homogeneous DE?

a) \(u(t)=u_1 (t)+⋯+c_{n+1} u_n (t)=∑_{k+1}^n=c_{k+1} u_k (t)\)

b) \(u(t)=u_1 (t)+⋯+u_n (t)=∑_{k=1}^nu_k(t) \)

c) \(u(t)=c_1 u_1 (t)+⋯+c_n u_n (t)=∑_{k=1}^n c_k u_k (t) \)

d) \(u(t)=c_0 u_0 (t)+⋯+c_n u_n (t)=∑_{k=0}^∞c_k u_k (t) \)

View Answer

Explanation: If we know n linearly independent solutions u

_{1}(t), …., u

_{n}(t) of a nth order linear homogeneous DE, then the general solution of this DE has the form:

\(u(t)=c_1 u_1 (t)+⋯+c_n u_n (t)=∑_{k=1}^nc_k u_k (t)\)

6. What is the degree of the homogeneous partial differential equation, \(\frac{∂^2 u}{∂t^2}-c^2 \frac{∂^2 u}{∂x^2}=0\)?

a) Second-degree

b) First-degree

c) Third-degree

d) Zero-degree

View Answer

Explanation: From the given equation, \(\frac{∂^2 u}{∂t^2}-c^2 \frac{∂^2 u}{∂x^2}=0\)we deduce that the power of the highest order term is 1, hence degree = 1.

7. The solution of an ODE contains arbitrary constants, the solution to a PDE contains arbitrary functions.

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.

8. Which of the following is not an example of linear differential equation?

a) y=mx+c

b) x+x’=0

c) x+x^{2}=0

d) x^”+2x=0

View Answer

Explanation: For a differential equation to be linear the dependent variable should be of first degree. Since in equation x+x

^{2}=0, x

^{2}is not a first power, it is not an example of linear differential equation.

9. A homogeneous linear differential equation has constant coefficients if it has the form

\(a_0 y+a_1 y’+a_2 y”+⋯+a_n y^{(n)}=0,\) where a_{1},…,a_{n} are (real or complex) numbers.

a) False

b) True

View Answer

Explanation: \(a_0 y+a_1 y’+a_2 y”+⋯+a_n y^{(n)}=0,\) is the form which represents a homogeneous linear differential equation which has constant coefficients. In other words, it has constant coefficients if it is defined by a linear operator with constant coefficients.

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. 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

12. 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.

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