Differential Equations and its Applications in Engineering Prof. (Dr.) Nita H. Shah Department of Mathematics, Gujarat University, Ahmedabad

Modeling with differential equations One of the most important application of calculus is differential equations, which often arise in describing some phenomenon in engineering, physical science etc. Parul Uni., Jan. 01, 2016

Concepts of differential equations In general, a differential equation is an equation that contains an unknown function and its derivatives. The order of a differential equation is the order of the highest derivative that occurs in the equation. A function y=f(x) is called a solution of a differential equation if the equation is satisfied when y=f(x) and its derivatives are substituted into the equation. Parul Uni., Jan. 01, 2016

Ordinary differential equations Definition: A differential equation is an equation containing an unknown function and its derivatives. Examples: 1. 2. 3. y is dependent variable and x is independent variable, and these are ordinary differential equations Parul Uni., Jan. 01, 2016

Partial differential equation Examples: 1. u is dependent variable and x and y are independent variables, and is partial differential equation. 2. 3. u is dependent variable and x and t are independent variables Parul Uni., Jan. 01, 2016

Order of Differential Equation The order of the differential equation is order of the highest derivative in the differential equation. Differential Equation ORDER 1 2 3 Parul Uni., Jan. 01, 2016

Degree of Differential Equation The degree of a differential equation is power of the highest order derivative term in the differential equation. Differential Equation Degree 1 1 3 Parul Uni., Jan. 01, 2016

Linear differential equation A differential equation is linear, if 1. dependent variable and its derivatives are of degree one, 2. coefficients of a term does not depend upon dependent variable. Example: 1. is linear. Example: 2. is non - linear because in 2nd term is not of degree one. Parul Uni., Jan. 01, 2016

3. 4. is non – linear Example: Example: is non - linear because in 2nd term coefficient depends on y. 3. Example: 4. is non - linear because is non – linear Parul Uni., Jan. 01, 2016

It is Ordinary/partial Differential equation of order… and of degree…, it is linear / non linear, with independent variable…, and dependent variable…. Parul Uni., Jan. 01, 2016

1st – order differential equation 1. Derivative form: 2. Differential form: . 3. General form: or Parul Uni., Jan. 01, 2016 Classfication of Differential Equation

First Order Ordinary Differential equation Parul Uni., Jan. 01, 2016

Second order Ordinary Differential Equation Parul Uni., Jan. 01, 2016 Classfication of Differential Equation

nth – order linear differential equation 1. nth – order linear differential equation with constant coefficients. 2. nth – order linear differential equation with variable coefficients Parul Uni., Jan. 01, 2016

Solution of Differential Equation Parul Uni., Jan. 01, 2016

Examples y=2x+c , is solution of the 1st order differential equation , c is arbitrary constant. As is solution of the differential equation for every value of c, hence it is known as general solution. Examples 1 parameter 2 parameters Parul Uni., Jan. 01, 2016

Families of Solutions Example Solution Observe that given any point (x0,y0), there is a unique solution curve of the above equation which curve goes through the given point. The solution is a family of ellipses.

Origin of differential equations solution Geometric Origin 1. For the family of straight lines the differential equation is 2. For the family of curves A. the differential equation is . B. the differential equation is Parul Uni., Jan. 01, 2016

g is acceleration due to gravity. Physical Origin 1. Free falling stone where s is distance or height and g is acceleration due to gravity. 2. Spring vertical displacement where y is displacement, m is mass and k is spring constant 3. RLC – circuit, Kirchoff ’s Second Law where q is charge on capacitor, L is inductance, c is capacitance, R is resistance and E is voltage Parul Uni., Jan. 01, 2016

Physical Origin Newton’s Low of Cooling where and its surrounding Ts is rate of cooling of the liquid, is temperature difference between the liquid ‘T’ and its surrounding Ts 2. Growth and Decay where y is the quantity present at any time Parul Uni., Jan. 01, 2016

Geometric point of view Geometrically, the general solution is a family of solution curves, which are called integral curves. When we impose an initial condition, we look at the family of solution curves and pick the one that passes through the point Physically, this corresponds to measuring the state of a system at time and using the solution of the initial-value problem to predict the future behavior of the system. Parul Uni., Jan. 01, 2016

Graphical approach: direction fields For most differential equations, it is impossible to find an explicit formula for the solution. Suppose we are asked to sketch the graph of the solution of the initial-value problem The equation tells us that the slope at any point (x,y) on the graph is f(x,y). To sketch the solution curve, we draw short line segments with slope f(x,y) at a number of points (x,y). The result is called a direction field. Parul Uni., Jan. 01, 2016

Example Ex. Draw a direction for the equation What can you say about the limiting value when Sol. Remark: equilibrium solution Parul Uni., Jan. 01, 2016

Separable equations Not all equations have an explicit formula for a solution. But some types of equations can be solved explicitly. Among others, separable equations is one type. A separable equation is a first-order differential equation in which the expression for can be factored into the product of a function of x and a function of y. That is, a separable equation can be written in the form Parul Uni., Jan. 01, 2016

Solutions of separable equations Thus a separable equation can be written into that is, the variables x and y are separated. We can then integrate both sides to get After we find the indefinite integrals, we get a relationship between x and y, in which there generally has an arbitrary constant. So the relationship determines a function y = y(x) and it is the general solution to the differential equation. Parul Uni., Jan. 01, 2016

Example Ex. Solve the differential equation Sol. Rewrite the equation into Integrate both sides which gives So, the general solution is Parul Uni., Jan. 01, 2016

Example Ex. Solve the differential equation Sol. Separate variables: Integrate: which is the general solution in implicit form. Remark: It is impossible to solve y in terms of x explicitly. Parul Uni., Jan. 01, 2016

Example Ex. Solve the differential equation Sol. C is arbitrary, but is not arbitrary. Verify that y=0 is also a solution. Therefore, where A is an arbitrary constant, is the general solution. Parul Uni., Jan. 01, 2016

Orthogonal trajectories An orthogonal trajectory of a family of curve is a curve that intersects each curve of the family orthogonally. For instance, each member of the family of straight lines is an orthogonal trajectory of the family To find orthogonal trajectories of a family of curve, first find the slope at any point on the family of curve, which is generally a differential equation. At any point on the orthogonal trajectories, the slope must be the negative reciprocal of the aforementioned slope. So the slope of orthogonal trajectories is governed by a differential equation, too. Let us solve the equation to get the orthogonal trajectories. Parul Uni., Jan. 01, 2016

Example Ex. Find the orthogonal trajectories of the family of curves where k is an arbitrary constant. Sol. Separating variables gives or Substituting into it, we find the slope at any point is At any point on orthogonal trajectory, the slope is Solving the equation, we get Parul Uni., Jan. 01, 2016

Example Ex. Suppose f is continuous and Find f(x). Sol. Parul Uni., Jan. 01, 2016

First order differential equations No general method of solutions of 1st O.D.E.s because of their different degrees of complexity. Possible to classify them as: exact equations equations in which the variables can be separated homogenous equations equations solvable by an integrating factor Parul Uni., Jan. 01, 2016

Exact equations Exact? General solution: F (x,y) = C For example Parul Uni., Jan. 01, 2016

Separable-variables equations In the most simple first order differential eqs., the independent variable and its differential can be separated from the dependent variable and its differential by the equality sign, using nothing more than the normal processes of elementary algebra. For example Parul Uni., Jan. 01, 2016

Homogeneous equations Homogeneous/nearly homogeneous? A differential equation of the type, Such an equation can be solved by making the substitution u = y/x and thereafter integrating the transformed equation. is termed a homogeneous differential equation of the first order. Parul Uni., Jan. 01, 2016

Homogeneous equation example Liquid benzene is to be chlorinated batchwise by sparging chlorine gas into a reaction kettle containing the benzene. If the reactor contains such an efficient agitator that all the chlorine which enters the reactor undergoes chemical reaction, and only the hydrogen chloride gas liberated escapes from the vessel, estimate how much chlorine must be added to give the maximum yield of monochlorbenzene. The reaction is assumed to take place isothermally at 55 C when the ratios of the specific reaction rate constants are: C6H6+Cl2  C6H5Cl +HCl C6H5Cl+Cl2  C6H4Cl2 + HCl C6H4Cl2 + Cl2  C6H3Cl3 + HCl k1 = 8 k2 ; k2 = 30 k3 Parul Uni., Jan. 01, 2016

Take a basis of 1 mole of benzene fed to the reactor and introduce the following variables to represent the stage of system at time , p = moles of chlorine present q = moles of benzene present r = moles of monochlorbenzene present s = moles of dichlorbenzene present t = moles of trichlorbenzene present Then q + r + s + t = 1 and the total amount of chlorine consumed is: y = r + 2s + 3t From the material balances : in - out = accumulation u = r/q Parul Uni., Jan. 01, 2016

Equations solved by integrating factor There exists a factor by which the equation can be multiplied so that the one side becomes a complete differential equation. The factor is called “the integrating factor”. where P and Q are functions of x only Assuming the integrating factor R is a function of x only, then is the integrating factor. Parul Uni., Jan. 01, 2016

Example Solve Let z = 1/y3 integral factor Parul Uni., Jan. 01, 2016

First order linear differential equations Summary of 1st O.D.E. First order linear differential equations occasionally arise in chemical engineering problems in the field of heat transfer, momentum transfer and mass transfer. Parul Uni., Jan. 01, 2016

First O.D.E. in heat transfer An elevated horizontal cylindrical tank 1 m diameter and 2 m long is insulated with asbestos lagging of thickness l = 4 cm, and is employed as a maturing vessel for a batch chemical process. Liquid at 950 C is charged into the tank and allowed to mature over 5 days. If the data below applies, calculated the final temperature of the liquid and give a plot of the liquid temperature as a function of time. Parul Uni., Jan. 01, 2016

Liquid film coefficient of heat transfer (h1) = 150 W/m2C Thermal conductivity of asbestos (k) = 0.2 W/m C Surface coefficient of heat transfer by convection and radiation (h2) = 10 W/m2C Density of liquid () = 103 kg/m3 Heat capacity of liquid (s) = 2500 J/kgC Atmospheric temperature at time of charging = 20 C Atmospheric temperature (t) t = 10 + 10 cos (/12) Time in hours () Heat loss through supports is negligible. The thermal capacity of the lagging can be ignored. Parul Uni., Jan. 01, 2016

Area of tank (A) = ( x 1 x 2) + 2 ( 1 / 4  x 12 ) = 2.5 m2 Rate of heat loss by liquid = h1 A (T - Tw) Rate of heat loss through lagging = kA/l (Tw - Ts) Rate of heat loss from the exposed surface of the lagging = h2 A (Ts - t) At steady state, the three rates are equal: Considering the thermal equilibrium of the liquid, input rate - output rate = accumulation rate t Ts Tw T B.C.  = 0 , T = 95 Parul Uni., Jan. 01, 2016

Second O.D.E. Purpose: reduce to 1st O.D.E. Likely to be reduced equations: Non-linear Equations where the dependent variable does not occur explicitly. Equations where the independent variable does not occur Homogeneous equations. Linear The coefficients in the equation are constant The coefficients are functions of the independent variable. Parul Uni., Jan. 01, 2016

They are solved by differentiation followed by the p-substitution. Non-linear 2nd O.D.E. - Equations where the dependent variables does not occur explicitly They are solved by differentiation followed by the p-substitution. When the p-substitution is made, the second derivative of y is replaced by the first derivative of p thus eliminating y completely and producing a first O.D.E. in p and x. Parul Uni., Jan. 01, 2016

Solve Let and therefore integral factor error function Parul Uni., Jan. 01, 2016

Non-linear 2nd O.D.E. - Equations where the independent variables does not occur explicitly They are solved by differentiation followed by the p-substitution. When the p-substitution is made in this case, the second derivative of y is replaced as Let Parul Uni., Jan. 01, 2016

Solve Let and therefore Separating the variables Parul Uni., Jan. 01, 2016

Non-linear 2nd O.D.E.- Homogeneous equations The homogeneous 1st O.D.E. was in the form: The corresponding dimensionless group containing the 2nd differential coefficient is In general, the dimensionless group containing the nth coefficient is The second order homogenous differential equation can be expressed in a form analogous to , viz. Assuming u = y/x Assuming x = et If in this form, called homogeneous 2nd ODE Parul Uni., Jan. 01, 2016

Solve Dividing by 2xy homogeneous Let Let Singular solution General solution Let Parul Uni., Jan. 01, 2016

Linear differential equations They are frequently encountered in most chemical engineering fields of study, ranging from heat, mass, and momentum transfer to applied chemical reaction kinetics. The general linear differential equation of the nth order having constant coefficients may be written: where (x) is any function of x. Parul Uni., Jan. 01, 2016

2nd order linear differential equations The general equation can be expressed in the form where P,Q, and R are constant coefficients Let the dependent variable y be replaced by the sum of the two new variables: y = u + v If v is a particular solution of the original differential equation The general solution of the linear differential equation will be the sum of a “complementary function” and a “particular solution”. Parul Uni., Jan. 01, 2016 purpose

The complementary function Let the solution assumed to be: auxiliary equation (characteristic equation) Unequal roots Equal roots Real roots Complex roots Parul Uni., Jan. 01, 2016

Unequal roots to auxiliary equation Let the roots of the auxiliary equation be distinct and of values m1 and m2. Therefore, the solutions of the auxiliary equation are: The most general solution will be If m1 and m2 are complex it is customary to replace the complex exponential functions with their equivalent trigonometric forms. Parul Uni., Jan. 01, 2016

Solve auxiliary function Parul Uni., Jan. 01, 2016

Equal roots to auxiliary equation Let the roots of the auxiliary equation equal and of value m1 = m2 = m. Therefore, the solution of the auxiliary equation is: Let where V is a function of x Parul Uni., Jan. 01, 2016

Solve auxiliary function Parul Uni., Jan. 01, 2016

Solve auxiliary function Parul Uni., Jan. 01, 2016

Particular integrals Two methods will be introduced to obtain the particular solution of a second linear O.D.E. The method of undetermined coefficients confined to linear equations with constant coefficients and particular form of  (x) The method of inverse operators general applicability Parul Uni., Jan. 01, 2016

Method of undetermined coefficients When  (x) is constant, say C, a particular integral of eq. When  (x) is a polynomial of the form where all the coefficients are constants. The form of a particular integral is When  (x) is of the form Terx, where T and r are constants. The form of a particular integral is Parul Uni., Jan. 01, 2016

Method of undetermined coefficients When  (x) is of the form G sin nx + H cos nx, where G and H are constants, the form of a particular solution is Modified procedure when a term in the particular integral duplicates a term in the complementary function. Parul Uni., Jan. 01, 2016

Solve auxiliary equation Equating coefficients of equal powers of x Parul Uni., Jan. 01, 2016

Method of inverse operators Sometimes, it is convenient to refer to the symbol “D” as the differential operator: But, Parul Uni., Jan. 01, 2016

The differential operator D can be treated as an ordinary algebraic quantity with certain limitations. (1) The distribution law: A(B+C) = AB + AC which applies to the differential operator D (2) The commutative law: AB = BA which does not in general apply to the differential operator D Dxy  xDy (D+1)(D+2)y = (D+2)(D+1)y Parul Uni., Jan. 01, 2016

The basic laws of algebra thus apply to the pure operators, The associative law: (AB)C = A(BC) which does not in general apply to the differential operator D D(Dy) = (DD)y D(xy) = (Dx)y + x(Dy) The basic laws of algebra thus apply to the pure operators, but the relative order of operators and variables must be maintained. Parul Uni., Jan. 01, 2016

Differential operator to exponentials More convenient! Parul Uni., Jan. 01, 2016

Differential operator to trigonometrical functions where “Im” represents the imaginary part of the function which follows it. Parul Uni., Jan. 01, 2016

Parul Uni., Jan. 01, 2016