2.1 Introduction

Consider the two-dimensional linear system

where a, b, c, d are real numbers. Depending on the nature of eigenvalues of the coefficient matrix , we can construct a non-singular real matrix C such that the reduces the system (2.1.1) into one of the following systems:

• (i) u′ = λu, v′ = μv

• (ii) u′ = λu + v, v′ = λv

• (iii) u′ = αu + βv, v′ = −βu + αv.

Here λ and μ are the eigenvalues of A, in case A has real eigenvalues; they may be equal. Whereas α and β(≠ 0) are respectively the real and the imaginary parts of the eigenvalues, in case A has non-real eigenvalues (they occur in conjugate pair) α ± iβ. It is easy to write down the solutions of the reduced systems. We have u(t) = u0eλt, v(t) = v0eμt for the case (i). The solutions are u(t) = (u0 + v0t)eλt, v(t) = v0eλt for the case (ii). Finally, for the case (iii), we have u(t) = eαt (u0 cos(βt) + v0 sin(βt)), v(t) = eαt(v0 cos(βt) − u0 sin(βt)). Using the matrix C, we can then write down the expressions for the solutions x, y.

The same procedure works for higher-dimensional linear systems. However, it is not easy to construct the matrix C as it involves deeper aspects of the eigenvalues and the corresponding eigenvectors of the coefficient matrix A and involves the Jordan canonical form. We only remark that each component of the solution of the linear system x′ = Ax is a linear combination of the products of the functions of t, and these functions are polynomials, exponentials and (in the presence of complex eigenvalues) the trigonometric functions–sine and cosine functions.

4.1 Introduction

A regular Sturm–Liouville system, or simply S–L system, refers to the following boundary value problem (BVP):

with prescribed boundary conditions at the end points a and b of the given interval [a, b]. Here p is a positive C1 function, q and ρ are continuous functions and ρ > 0, all defined on a bounded interval [a, b], and λ is a real parameter. When p vanishes somewhere in the interval or the interval under consideration is unbounded, the problem is termed as singular. Singular BVPs are more difficult to deal with. The interested reader may refer to Ref. [15].

It turns out that the non-trivial solutions to the BVP exist only for a discrete set of values of the parameter λ, tending to infinity. The situation may thus be compared with the eigenvalues of a matrix, considered as a linear operator on a finite-dimensional space. The main difference is that we are now working in an infinite dimensional space. In analogy with matrices, the discrete set of the values of the parameter for which the non-trivial solutions exist is called the eigenvalues of the BVP, and the corresponding non-trivial solutions are called the eigenfunctions.

Again, continuing with the similarity with matrices, we know that any vector in ℝn can be written as a unique linear combination of eigenvectors of a real, symmetric matrix. In this case, the eigenvectors are also orthogonal; we discuss more regarding orthogonality in the next section. Surprisingly, these concepts can be extended to the operator. In the case of a BVP too, one may express an arbitrary function as a linear combination of the corresponding eigenfunctions. Since these are (infinite) power series, we need to discuss the convergence and so on, and the tools required come from functional analysis (Hilbert space).

The S–L systems arise in many physical problems, for example, in the following situation. Consider the longitudinal vibrations of an elastic bar of local stiffness ρ(ξ) and density ρ(ξ).

1.1 Introduction

The first-order equations–linear and non-linear–and the second-order linear equations, with constant or variable coefficients, are considered in this chapter. For solving the first-order equations, familiar methods such as the method of separation of variables or a method that can be reduced to this are used. Also discussed are the exact differential equations or those equations that can be reduced to this form using a suitable integrating factor (IF). We also emphasize the peculiarities that may arise in an initial value problem (IVP) when sufficient conditions imposed in the Cauchy–Peano existence theorem or in the method of Picard's iterations are not satisfied. Many exercises deal with the maximal interval of existence of a solution to an IVP.

Only second-order linear equations are considered here. The non-linear equations or, more generally, the two-dimensional systems of first-order equations are treated in Chapter 5 on qualitative analysis. The treatment of equations with constant coefficients is straightforward. The equations with variable coefficients are more difficult to deal with, and, in general, it is not possible to obtain the solution in explicit form. However, the structure of solutions to the homogeneous and inhomogeneous equations is well-understood.

A general first-order ordinary differential equation (ODE) takes the form f(t, x(t), x′(t)) = 0, where f is a given function and x = x(t) is the unknown function to be determined. A general theory for the above equation is rather difficult.

The present book, consisting of two parts, is essentially a collection of Exercises and their Solutions in Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). Its special nature, which makes it different from the usual books on the subject, consists of providing sufficient theoretical materials at the beginning of each chapter, pertaining to that chapter. This certainly helps students straight way dwell into solving the exercises, referring to the relevant textbooks only for understanding the proofs of the important theorems and additional topics. The topics are at (senior) undergraduate and beginning graduate levels.

The importance of the subjects of ODE and PDE in the modern science and engineering curricula is well-known. Many interesting and important real-life problems are modelled using ODE and PDE. These include, but not limited to, physics, chemistry, biology, engineering, economics, sociology and psychology. These topics require extensive mathematical techniques, which are dealt with in many existing books on the subjects, with varying degrees of difficulty and contents. Many books, especially those used as textbooks, also contain a comprehensive list of exercises. Certain books also have Solutions Manuals, usually made available to teachers only. Many other books contain solutions to a selected list of exercises. In our experience, as teachers of these subjects for many years, we find that many students do not attempt to solve the exercises, except routine straightforward ones. This could be partially due to the fact that some exercises turn out to be tricky or difficult, often requiring some advanced knowledge in analysis and linear algebra for their solutions. Furthermore, those students who study the subjects on their own find solving exercises more difficult, thus missing out an important aspect of learning the subject.

This book is an outcome of our experience of teaching and research, not only in our respective institutions but also in many other institutions and universities across India. We have extensively given lectures on differential equations at these institutions, for more than three decades, as part of workshops, schools and series of lectures. After the successful publication of our two co-authored books, one on ODEs (2017) (Ref.[46]) and the other on PDEs (2020) (Ref.[45]), there was a constant inquiry about the Solutions Manuals of the problems in these books from the student and teacher community, as well as our well-wishers.

M. Vijayanunni, the former census commissioner, continued to speak out publicly against the combined caste-wise enumeration and BPL survey just before the project began in late July 2011. His prescient comments outlined the legal distinction between the decennial census and the BPL survey, related operational differences, and the shortcomings of the revised survey instrument. Vijayanunni pointed out that the SEC survey remained a project designed to identify those households living below the poverty line and differentiate among households that faced different degrees of food insecurity, housing insecurity, and poverty. While the project sought to pinpoint households above the poverty line, it did not seek to distinguish within this broad grouping. Detailing the socioeconomic profiles of comparatively privileged households fell outside the scope of the BPL identification. As such, degrees of privilege remained comparatively masked by the survey instrument. The survey did not seek to generate “a caste-wise socio-economic profile of the population of India” as required for the implementation of reservation benefits and related legal cases. Vijayanunni argued that inserting a caste question into a survey whose purpose was to identify poor households could not generate a comprehensive caste-wise socioeconomic profile of the country, nor a more detailed understanding of caste-based power.

We are standing in front of a three-floor apartment building in central Bengaluru. “Thirty-seven,” says Vijaya as she locates the building on the block map and shares the number from the map associated with the building. She folds the map and places it under the household listing booklet on her clipboard. Vijaya has been trained as an enumerator for the SEC survey and she and her husband, Mohan, who is the DEO, are getting ready to interview their first household of the afternoon. She opens the household listing booklet and flips through it until she reaches the three entries for building number 37. She locates the number associated with the ground floor unit and tells her husband, “148.” Mohan types the number into his handheld tablet. The tablet contains preloaded data from the NPR, a recently created government database for a system of national identification. A few seconds later the data for household 148 populates the screen. Mohan reads aloud the name of the head of household, “R.L. Suresh.” Vijaya nods to indicate that “R.L. Suresh” matches the handwritten entry for the head of household in the household listing booklet, which census enumerators created during the first round of Census 2011 two years earlier. Mohan opens the front gate, and we walk into the small compound and up to the front door of the ground-floor unit. Mohan knocks on the front door with his right hand and holds the tablet in his left hand. A few seconds later, a woman in her mid- to late thirties opens the door and looks at the three of us. Mohan holds up the handheld device as he explains that we are here “for census work.” He then asks, “Is this the house of R.L. Suresh?” The woman nods in acknowledgment. She invites us into the drawing room and asks us to sit. As we settle into chairs, she closes the front door and stands near the entrance. Her teenage son stands in a corner of the room and watches. Mohan confirms the name, sex, and age of the four household members based on preloaded NPR data.

In their 2008 article “A Sociology of Quantification,” Wendy Espeland and Mitchell Stevens describe the immense bureaucratic effort required to quantify populations:

Chapters 3–6 trace the extensive infrastructure involved in the production of caste-wise data. In response to organizing by caste census advocates, the Congress-led government conceded to collect caste-wise data and then mobilized massive financial, human, and technological resources to do so. This infrastructure included, but was not limited to, executive bureaucrats who blocked the caste-wise enumeration in the census and pushed it into a BPL survey; MoRD administrators in Delhi who had been revamping the BPL survey for several years, who collaborated with the MoHUPA and the ORGI to coordinate and manage the combined BPL survey and caste-wise enumeration; state and local government officials that oversaw the implementation of the extensive ground operations for the survey; PSUs that managed electronic data entry operations; private firms that bid for, and secured, PSU-managed government contracts to staff and implement electronic data entry; charge centers—or the local hub for data collection operations, where data collectors uploaded interview data to a government server—and the people and technology that the staffed charge centers across the country; the government server hosted by the National Informatics Centre (NIC) that stored all collected household data; the entire training infrastructure for data collectors which prepared them to conduct household interviews; the hundreds of thousands of enumerators and DEOs who interviewed households, as well as the supervisors who monitored their work;

9.1 Homogeneous Equation: Fourier–Poisson Formula

Consider the initial value problem (IVP) for the heat equation in space dimension n ⩾ 1:

The solution of the IVP (9.1.1) is given by the Fourier–Poisson formula:

for x ∈ ℝn and t > 0. The Fourier–Poisson formula (9.1.2) can be written as a convolution. For this purpose, we define the heat kernel or fundamental solution of the heat equation by

We can write the Fourier–Poisson integral (9.1.2) as the convolution of K and g as. Furthermore, the heat kernel K enjoys the following properties, which are frequently used:

9.1.1 Inhomogeneous Equation: Duhamel's Principle

We now consider the inhomogeneous heat equation:

Assume that the function f and its partial derivatives fxj are continuous in x ∈ ℝn, t > 0 and the function g is continuous and bounded. Owing to the linearity in the problem, the required solution can be written as some of two functions, namely the solution of the homogeneous equation with initial condition g and the solution of the inhomogeneous equation with zero initial condition. Thus, it suffices to consider equation (9.1.4) with g ≡ 0 therein. A solution of this problem can be obtained via the Duhamel's principle: Fix s ⩾ 0, and consider the IVP for the heat equation:

Denoting the solution by v(x, t; s), we get

using equation (9.1.2) and the change of variable t ↦ t − s. We can now write down the expression for a solution to equation (9.1.4) with g ≡ 0, as:

9.1.2 Heat Equation in a Finite Interval: Fourier Method

In this subsection, we consider initial boundary value problem for the one-dimensional heat equation and see how the Fourier method can be used to obtain the solution. Consider IVP for the heat equation in a finite interval [0, L] on the real line

The initial condition u(x, 0) = g(x) in equation (9.1.8) represents the initial temperature distribution at all points of the rod at the initial instant of time t = 0. At the end points, x = 0 and x = L, different boundary conditions may be given.

Appendix A: Socio-Economic Caste (SEC) Survey Questionnaire (Urban)

For Each Household

Section 1: Housing/Dwelling

• 1. Predominant material of wall of the dwelling room (s) (Give code)

  • 1 = Grass/thatch/bamboo, etc.

  • 2 = Plastic/polythene

  • 3 = Mud/unburnt brick

  • 4 = Wood

  • 5 = Stone not packed with mortar

  • 6 = Stone packed with mortar

  • 7 = G.I./metal/asbestos sheets

  • 8 = Burnt brick

  • 9 = Concrete

  • 0 = Any other

• 2. Predominant material of roof of the dwelling room (s) (Give code)

  • 1 = Grass/thatch/bamboo/wood/mud, etc.

  • 2 = Plastic/polythene

  • 3 = Hand-made tiles

  • 4 = Machine-made tile

  • 5 = Burnt brick

  • 6 = Stone

  • 7 = Slate

  • 8 = G.I./metal/asbestos sheets

  • 9 = Concrete

  • 0 = Any other

• 3. Ownership status of this house (Give code)

  • 1 = Owned

  • 2 = Rented

  • 3 = Shared

  • 4 = Living on premises with employer

  • 5 = House provided by employer

  • 6 = Any other

• 4. Number of dwelling rooms exclusively in possession of this household (Record 1,2,3…)

Section 2: Amenities

• 5. Availability of drinking water source

  • 1 = Within the premises, 2 = Near the premises, 3 = Away

• 6. Main source of lighting

  • 1 = Electricity, 2 = Kerosene, 3 = Solar, 4 = Other oil, 5 = Any other, 6 = No lighting

• 7. Water-seal latrine exclusively for the household

  • 1 = Yes, 2 = No

• 8. Wastewater outlet connected to

  • 1 = Closed drainage, 2 = Open drainage, 3 = No drainage

• 9. Separate room used as kitchen exclusively for the household

  • 1 = Yes, 2 = No

Section 3: Assets

Does the household own the following assets (Give code)

• 10. Refrigerator

  • 1 = Yes; 2 = No

• 11. Telephone/Mobile phone

  • Yes: 1 = Landline only, 2 = Mobile only, 3 = Both, 4 = No

• 12. Computer/Laptop

  • Yes: 1 = With internet, 2 = Without internet, 3 = No

• 13. Motorized wheelers

  • 1 = Two/Three-wheeler, 2 = Four-wheeler, 3 = No

• 14. AC

  • 1 = Yes; 2 = No

• 15. Washing machine

  • 1 = Yes; 2 = No

For Each Person

• 1. Serial No.

7.1 Introduction

In many branches of mathematics, we do classify the objects of study. For example, in linear algebra, we classify the square matrices into symmetric matrices, orthogonal matrices and others. Similarly, we may group a particular set of partial differential equations (PDE) whose solutions have common qualitative and/or quantitative properties. Among the important equations of mathematical physics, the three main equations, namely the wave equation, Laplace equation and the heat equation, are classified as follows:.

Wave Equation: This equation is a prototype of hyperbolic equations. In one dimension, this equation models many real-world problems, such as small transversal vibrations of a string, longitudinal vibrations of a rod, electrical oscillations in a wire, torsional oscillations of shafts and oscillations in gases.

In two dimensions, the wave equation models vibrations of a membrane, and in three dimensions, it models the propagation of sound waves, light waves and electromagnetic waves in a medium.

Heat Equation: It models the heat diffusion in a medium and is a prototype of parabolic equations. It also describes the diffusion of a chemical substance in a medium. As such, it is also called diffusion equation.

Laplace Equation: A steady-state process, that is, a physical process that does not change with time, is generally modelled by a Laplace or Poisson equation and is classified as an elliptic equation. A physical process that is in equilibrium state is also described by an elliptic equation.

Another reason for the classification of PDE is more mathematical in nature by the consideration of a comparison with ordinary differential equation (ODE). Typically, an ODE is associated with a Cauchy problem or an IVP.

5.1 Introduction

The stability analysis of equilibrium points of an autonomous first-order system

is an important topic in the qualitative theory of ordinary differential equations (ODE). Here x = x(t) ∈ ℝn is a vector valued unknown function of the independent variable t ∈ I, an interval in ℝ, and f: ℝn → ℝn is a given vector valued function, which is assumed to be a C1 or more smooth function. This assumption ensures the uniqueness (local or global) of a solution of the system (5.1.1) with a prescribed initial value at an initial time. The positive integer n is referred to as the dimension of the system.

The system (5.1.1) is called an autonomous system because the right-side function f does not depend on t explicitly. When f depends on t explicitly as well, the system is referred to as non-autonomous. For example, the equation x′ = x + t (1D or one-dimensional equation) is non-autonomous.

In some situations, we do assume more smoothness on f, so that global existence of a solution is guaranteed; this means existence for all t. Even when a solution does not exist for all t, we can still do the phase space analysis by considering the maximum interval of existence of the solution in question. However, uniqueness plays a crucial role. In what follows, we introduce many concepts, definitions and list many results that are useful in solving the exercises. For proofs and other details, we refer to Ref.[46] or any other book with similar contents.

On July 3, 2015, four years after the MoRD started data collection for the SEC survey in the state of Tripura, it co-announced the release of provisional data for rural India. Finance minister Arun Jaitley and minister of rural development Chaudhary Birendra Singh—both of the BJP-led government that had come to power in 2014—published a press statement that included a link to SEC survey summary data on the MoRD website. The MoRD had analyzed and made public the data that it required for BPL identification, and envisioned numerous additional possibilities for using the collected data. The press release stated that the MoRD planned “to use the [SEC survey] data in all its programmes,” specifically citing “Housing for all, Education and Skills thrust, MGNREGA [Mahatma Gandhi National Rural Employment Guarantee Act], National Food Security Act, interventions for differently able[d], interventions for women-led households, and targeting of households/ individual entitlements on evidence of deprivation, etc.” It also pointed to other intended uses of the SEC survey data by policy planners across levels of government and stated that combining the SEC survey data with the NPR would allow coordination across programs “to simultaneously address the multi-dimensionality of poverty by addressing the deprivation of households in education, skills, housing, employment, health, nutrition, water, sanitation, social and gender mobilization and entitlement” and tracking the progress of households over time. The concluding paragraph of the press release argued that the “[SEC survey] truly makes evidence based targeted household interventions for poverty reduction possible.” The two summary tables in the press release included details on the percentage of households automatically included on the BPL list based on five possible parameters for automatic inclusion, one of which was “whether any member of the household was from a primitive tribal group” (a question from the household section of the SEC survey questionnaire). This data came from a question that had been field-tested in the 2010 pilot survey—so it was not an addition from when the caste-wise enumeration was combined with the BPL survey. The tables also included the percentage of SC and ST households, which previous BPL surveys had documented.