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## Calculus - Introduction to Differential Equations and Solved Problems - Outline of Contents:

### Target Audience: High School Students, College Freshmen and Sophomores, students preparing for the International Baccalaureate (IB), AP Calculus AB, AP Calculus BC, A Level, Singapore/GCE A-Level; Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE/AIEEE Anyone else who needs this Tutorial as a reference!

Theory and definitions. What a differential equation is; ordinary and partial differential equations; order and degree of a differential equation; linear and non linear differential equations; General, particular and singular solutions; Initial and boundary value problems; Linear independence and dependence; Homogeneous equations; First order differential equations; Characteristic and auxiliary equations. Introductory problems demonstrating these concepts. Introducing the concept of Integrating Factor (IF).

## Here's a quick outline of what this tutorial introduces :

### • Diﬀerential Equation:

A diﬀerential equation can be deﬁned as an equation containing derivatives of various orders and the variables.

### • Ordinary Diﬀerential Equation:

Diﬀerential equations which involve one independent variable are called ordinary diﬀerential equations.

### • Partial Diﬀerential Equation:

If the diﬀerential equation involves more than one independent variable and partial derivatives of the dependent variable with respect to them, then it is called a partial diﬀerential equation.

### • Order and Degree Of a Diﬀerential Equation:

The order of the highest order derivative involved in a diﬀerential equation is called the order of the diﬀerential equation. The degree of a diﬀerential equation is the degree of the highest derivative which occurs in it, after the diﬀerential equation has been made free from radicals and fractions as far the derivatives are concerned.

### • Linear and Non-Linear Diﬀerential Equations:

A diﬀerential equation is called linear if dependent variable and its diﬀerential co-eﬃcients occur in the ﬁrst degree only. The coeﬃcients of dependent variable and its derivatives may be any functions independent variable.

### • General,Particular and Singular Solutions:

A particular solution is any one solution of the diﬀerential equation. The general solution of a diﬀerential equation is the set of all particular solutions. A solution of a diﬀerential equation, which cannot be obtained from its general solution by assigning any particular values to the arbitrary constants, is called a singular solution.

### • Initial-Value Problem(IVP) and Boundary-Value Problem(BVP):

A diﬀerential equation together with given initial conditions on the dependent variable and its derivatives at the same value of the independent variable, constitute an initial-value problem. A diﬀerential equation with boundary conditions at more than one value of the in- dependent variable, forms a boundary-value problem.

### • Linear Independence and Dependence:

Let f1 (x), f2 (x), . . . , fn (x) be n functions. Then, these functions are said to be linearly independent on some interval I, if the equation
c1 f1 (x) + c2 f2 (x) + . . . + cn fn (x) = 0 implies c1 = 0 = c2 = . . . = cn .
These functions are said to be linearly dependent on I, if (1) holds for c1 , c2 , . . . , cn not all zero.

### • Homogeneous Equations:

A diﬀerential equation
dy/dx = f1 (x, y) / f2 (x, y)
where f1 and f2 are homogeneous functions of x and y of the same degree, is called a homogeneous diﬀerential equation. Such equations can be solved by taking a new dependent variable v connected with the old one y by the equation y = vx. The variables are made separable which makes it easier to solve the equations.

### • Characteristic/Auxiliary equation:

Consider the linear homogeneous second order equationay + by + cy = 0, a, b, c are constants. In operator notation, (aD2 + bD + c)y = 0. For a particular solution we try to find the form y = emx , where m is an unknown constant to be determined. Then we get an algebraic equation in m. It is called the characteristic equation or the auxiliary equation of the linear homogeneous equation.

### In Case you'd like to take a look at some of our other tutorials related to Single Variable Calculus :

 Quick and introductory definitions related to Funtions, Limits and Continuity - Defining the domain and range of a function, the meaning of continuity, limits, left and right hand limits, properties of limits and the "lim" operator; some common limits;  defining the L'Hospital rule, intermediate and extreme value theorems. Functions, Limits and Continuity - Solved Problem Set I - The Domain, Range, Plots and Graphs of Functions;  L'Hospital's Rule- -  Solved problems demonstrating how to compute the domain and range of functions, drawing the graphs of functions, the mod function, deciding if a function is invertible or not; calculating limits for some elementary examples, solving 0/0 forms, applying L'Hospital rule. Functions, Limits and Continuity - Solved Problem Set II - Conditions for Continuity, More Limits, Approximations for ln (1+x) and sin x for infinitesimal values of x   - More advanced cases of evaluating limits, conditions for continuity of functions, common approximations used while evaluating limits for ln ( 1 + x ), sin (x); continuity related problems for more advanced functions than the ones in the first group of problems (in the last tutorial). Functions, Limits and Continuity - Solved Problem Set III - Continuity and Intermediate Value Theorems - Problems related to Continuity, intermediate value theorem. Introductory concepts and definitions related to Differentiation - Basic formulas, Successive Differentiation, Leibnitz, Rolle and Lagrange Theorems, Maxima , Minima, Convexity, Concavity, etc - Theory and definitions introducing differentiability, basic differentiation formulas of common algebraic and trigonometric functions , successive differentiation, Leibnitz Theorem, Rolle's Theorem,  Lagrange's Mean Value Theorem, Increasing and decreasing functions, Maxima and Minima; Concavity, convexity and inflexion, implicit differentiation. Differential Calculus - Solved Problem Set I - Common Exponential, Log , trigonometric and polynomial functions - Examples and solved problems - differentiation of common algebraic, exponential, logarithmic, trigonometric and polynomial functions and terms; problems related to differentiability . Differential Calculus - Solved Problem Set II - Derivability and continuity of functins - Change of Indepndent Variables - Finding N-th Derivatives - Examples and solved problems - related to derivability and continuity of functions; changing the independent variable in a differential equation; finding the N-th derivative of functions Differential Calculus - Solved Problems Set III- Maximia, Minima, Extreme Values,  Rolle's Theorem - Examples and solved problems - related to increasing and decreasing functions; maxima, minima and extreme values; Rolle's Theorem Differential Calculus - Solved Problems Set IV - Points of Inflexion, Radius of Curvature, Curve Sketching -  Examples and solved problems - Slope of tangents to a curve, points of inflexion, convexity and concavity of curves, radius of curvature and asymptotes of curves, sketching curves Differential Calculus - Solved Problems Set V - Curve Sketching, Parametric Curves - More examples of investigating and sketching curves, parametric representation of curves Introducing Integral Calculus - Definite and Indefinite Integrals - using Substitution , Integration By Parts, ILATE rule  - Theory and definitions. What integration means, the integral and the integrand. Indefinite integrals, integrals of common functions.  Definite integration and properties of definite integrals; Integration by  substitution, integration by parts, the LIATE rule, Integral as the limit of a sum. Important forms encountered in integration. Integral Calculus - Solved Problems Set I - Basic examples of polynomials and trigonometric functions, area under curves - Examples and solved problems - elementary examples of integration involving trigonometric functions, polynomials; integration by parts; area under curves. Integral Calculus - Solved Problems Set II - More integrals, functions involving trigonometric and inverse trigonometric ratios - Examples and solved problems - integration by substitution, definite integrals, integration involving trigonometric and inverse trigonometric ratios. Integral Calculus - Solved Problems Set III - Reduction Formulas, Using Partial FractionsI- Examples and solved problems - Reduction formulas, reducing the integrand to partial fractions, more of definite integrals Integral Calculus - Solved Problems Set IV - More of integration using partial fractions, more complex substitutions and transformations - Examples and solved problems - More of integrals involving partial fractions, more complex substitutions and transformations Integral Calculus - Solved Problems Set V- Integration as a summation of a series - Examples and solved problems - More complex examples of integration, examples of integration as the limit of a summation of a series Introduction to Differential Equations and Solved Problems - Set I - Order and Degree, Linear and Non-Linear Differential Equations, Homogeneous Equations, Integrating Factor -  Theory and definitions. What a differential equation is; ordinary and partial differential equations; order and degree of a differential equation; linear and non linear differential equations; General, particular and singular solutions; Initial and boundary value problems; Linear independence and dependence; Homogeneous equations; First order differential equations; Characteristic and auxiliary equations. Introductory problems demonstrating these concepts. Introducing the concept of Integrating Factor (IF). Differential Equations - Solved Problems - Set II - D operator, auxillary equation, General Solution - Examples and solved problems - Solving linear differential equations, the D operator, auxiliary equations. Finding the general solution ( CF + PI ) Differential Equations - Solved Problems - Set III - More Differential Equations - More complex cases of differential equations. Differential Equations - Solved Problems - Set IV - Still more differential equations.