# Partial Differential Equations Questions and Answers – Non-Homogeneous Linear PDE with Constant Coefficient

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This set of Partial Differential Equations Questions and Answers for Experienced people focuses on “Non-Homogeneous Linear PDE with Constant Coefficient”.

1. Non-homogeneous which may contain terms which only depend on the independent variable.
a) True
b) False

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 a non-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) $$\frac{∂u}{∂t}-T \frac{∂^2 u}{∂x^2}=0$$

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).

3. What is the general form of the general solution of a non-homogeneous DE (uh(t)= general solution of the homogeneous equation, up(t)= any particular solution of the non-homogeneous equation)?
a) u(t)=uh (t)/up (t)
b) u(t)=uh (t)*up (t)
c) u(t)=uh (t)+up (t)
d) u(t)=uh (t)-up (t)

Explanation: The general solution of an inhomogeneous ODE has the general form: u(t)=uh (t)+up (t), where uh (t) is the general solution of the homogeneous equation, up (t) is any particular solution of the non-homogeneous equation.

4. While an ODE of order m has m linearly independent solutions, a PDE has infinitely many.
a) False
b) True

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.

5. Which of the following methods is not used in solving non-homogeneous equations?
a) Exponential Response Formula
b) Method of Undetermined Coefficients
c) Orthogonal Method
d) Variation of Constants

Explanation: There are several methods in solving a non-homogeneous equation. Some of them are:

• Exponential Response Formula
• Method of Undetermined Coefficients
• Variation of Constants
• annihilator method

6. What is the order of the non-homogeneous partial differential equation,
$$\frac{∂^2 u}{∂x^2}+(\frac{∂^2 u}{∂x∂y})^2+\frac{∂^2 u}{∂y^2}=x^2+y^2$$?
a) Order-3
b) Order-2
c) Order-0
d) Order-1

Explanation: The order of an equation is defined as the highest order derivative present in the equation and hence from the equation, $$\frac{∂^2 u}{∂x^2}+(\frac{∂^2 u}{∂x∂y})^2+\frac{∂^2 u}{∂y^2}=x^2+y^2$$, it is clear that it of 2nd order.

7. What is the degree of the non-homogeneous partial differential equation,
$$(\frac{∂^2 u}{∂x∂y})^5+\frac{∂^2 u}{∂y^2}+\frac{∂u}{∂x}=x^2-y^3$$?
a) Degree-2
b) Degree-1
c) Degree-0
d) Degree-5

Explanation: Degree of an equation is defined as the power of the highest derivative present in the equation. Hence from the equation,$$(\frac{∂^2 u}{∂x∂y})^5+\frac{∂^2 u}{∂y^2}+\frac{∂u}{∂x}=x^2-y^3$$, the degree is 5.

8. The Integrating factor of a differential equation is also called the primitive.
a) True
b) False

Explanation: The general solution of a differential equation is also called the primitive. The solution of a partial differential equation obtained by eliminating the arbitrary constants is called a general solution.

9. A particular solution for an equation is derived by substituting particular values to the arbitrary constants in the complete solution.
a) True
b) False

Explanation: A solution which does not contain any arbitrary constants is called a general solution whereas a particular solution is derived by substituting particular values to the arbitrary constants in this solution.

10. What is the complete solution of the equation, $$q= e^\frac{-p}{α}$$?
a) $$z=ae^\frac{-a}{α} y$$
b) $$z=x+e^\frac{-a}{α} y$$
c) $$z=ax+e^\frac{-a}{α} y+c$$
d) $$z=e^\frac{-a}{α} y$$

Explanation: Given: $$q= e^\frac{-p}{α}$$
The given equation does not contain x, y and z explicitly.
Setting p = a and q = b in the equation, we get $$b= e^\frac{-p}{α}.$$
Hence, a complete solution of the given equation is,
z=ax+by+c, with $$b= e^\frac{-a}{α}$$
$$z=ax+e^\frac{-a}{α} y+c.$$

11. In recurrence relation, each further term of a sequence or array is defined as a function of its succeeding terms.
a) True
b) False

Explanation: An equation that gives a sequence such that each next term of the sequence or array is defined as a function of its preceding terms, is called a recurrence relation.

12. What is the degree of the differential equation, x3-6x3 y3+2xy=0?
a) 3
b) 5
c) 6
d) 8 