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- 1. MAT 435 CHAPTER 4 : DIFFERENTIAL EQUATIONS 4.1: INTRODUCTION Definition: An equation containing the derivatives of one or more dependent variables with respect to one or more independent variables, is said to be a differential equation (DE). A. CLASSIFICATION BY TYPE If an equation contains only ordinary derivatives of one or more variables with respect to a single independent variable is said to be an ordinary differential equation (ODE). dy + 5y = e x dx d2 y ODE dy + 6y = 0 dx dx dx dy + = 2x + y dt dt 2 − ODE ODE An equation involving partial derivatives of one or more dependent variables of two or more independent variables is called a partial differential equation (PDE). ∂ 2u ∂x 2 ∂ 2u 2 + = ∂ 2u ∂y 2 ∂ 2u 2 ∂x ∂t ∂u ∂V =− ∂y ∂x =0 −2 PDE ∂u ∂t PDE PDE Ordinary derivatives writing notation: a) Leibniz notation: 1
- 2. dy dx d2 y dx 2 d3 y dx 3 ( y’ , y’’ , y’’’ , y b) Prime notation: 4) … c) Newton’s dot notation / flyspeck notation: d2S Example: dt 2 = − 32 ⇒ •• S = − 32 U xx = U tt − 2U t d) Partial derivatives in subscript notation: B. CLASSIFICATION BY ORDER The order of a differential equation (either ODE or PDE) is the order of the highest derivatives in the equation. 3 d2 y dy + 5 − 4y = e x 2 dx dx Second order ODE M(x, y)dx + N(x, y)dy = 0 First order ODE ( y – x )dx + 4xdy = 0 First order ODE C. CLASSIFICATION BY LINEARITY ( F(x , y , y’ , y’’ , y’’’ , … y n) )=0 ( An nth order ODE is said to be linear if F is linear in y , y’ , y’’ , y’’’ , … y n) . This means that th an n order ODE is when ( ( a n x )y n ) + a n −1 x )y n −1) + ... + a 1 x )y '+a 0 x )y − g x ) = 0 ( ( ( ( ( an x ) ( dn y dx n + a n −1 x ) ( d n −1y dx n −1 + ... + a1 x ) ( dy + a 0 x )y − g x ) = 0 ( ( dx 2 or
- 3. Linear first order DE a1 x ) ( dy + a 0 x )y = g x ) ( ( dx Linear second order DE a2 x ) ( d2 y dx 2 + a1 x ) ( dy + a 0 x )y = g x ) ( ( dx Example: (y – x) dx + 4x dy = 0 linear, 1st order, ODE Y” – 2y’ + y = 0 linear, 2nd order, ODE d3 y dx 3 +x dy − 5y = e x dx linear, 3rd order, ODE Non-linear ODE (1 – y) y’ + 2y = e x d2 y dx 2 d4 y dx 4 non-linear, 1st order, ODE + sin y = 0 non-linear, 2nd order, ODE + y2 = 0 non-linear, 4th order, ODE Solution of an ODE Any function φ, defined on an interval I and possessing at least n derivatives that are continuous on I, which when substituted into an nth order ordinary differential equation reduces the equation to an identity, is said to be a solution of the equation on the interval. Example 4.1.1 3
- 4. State the order of the given ordinary differential equation. Determine whether the equation is linear or non-linear. a) (1 – x) y’ – 4xy = cos x b) c) t 4 y( 5 ) − ty " + 6y = 0 d2 y dy = 1+ 2 dx dx 2 d) (sin θ)y ' ' ' − (cos θ)y ' = 2e y D. TERMINOLOGY Differential equation – An equation containing the derivatives of differentials of one or more dependent variables with respect to one or more independent variables. Ordinary differential equation – An equation which contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable. Partial differential equation – An equation involving the partial derivatives of one or more dependent variables of two or more independent variables. The order of differential equation – The order of the highest derivatives in a DE. Solution – A solution of a DE is any function which satisfies the equation, i.e the solution reduces DE to an identity. Initial value problem – The unknown function x and its derivative are specified at one value of the independent variable at t = 0. 4
- 5. Boundary value problem – The unknown function x are specified at two values of the independent variables, example at t = 0 and at t = 1. 4.2: FIRST ORDER DIFFERENTIAL EQUATION There are four standard types of first order differential equations: a) Separable equations b) Homogeneous equations c) Exact equations d) Linear equations A. SEPARABLE VARIABLES EQUATIONS dy =g x) ( dx Standard form: h y) ( Can be written in differential form: h y )dy = g x )dx ( ( The general solution is then obtained directly by integration on both sides. h ∫( Which equivalent to: H(y) = G(x) + C Example 4.2.1 Solve the differential equation. a) dy = −4 xy 2 dx b) ( x + 1) c) dy = 1− y dx y x + 3)+ x 3 − y ) ( ( dy =0 dx 5 ∫ y )dy = g x )dx (
- 6. Example 4.2.2 Solve the equation. a) ( 1 + cos θ) dr = r sin θ dθ b) dy = dx xy x2 − 4 Example 4.2.3 Solve the initial value problem. a) ( 4y − cos y ) dy − 3x 2 = 0 dx b) dy = xy 2 e x dx c) dy 3 x 2 + 4 x + 2 = dx 2 y − 1) ( ; ; y 0) = 0 ( y 0) = 2 ( ; y 0) = 1 ( B. HOMOGENEOUS EQUATIONS Standard equation: dy y = g dx x or x dy = G y dx y Where g are functions in terms of x Use substitution : x and G are functions in terms of y y = vx dy dv =v+x dx dx To transform the standard equation into separable equation. 6 .
- 7. Theorem: If M(x, y) dx + N(x, y) dy = 0 is a Homogeneous Equation, then the change of variables y = vx, transforms it into a Separable Equation in the variables v and x. Example 4.2.4 dy = y2 Solve the equation, x 2 + xy dx . Example 4.2.5 Solve the equation,( x 2 − 3y 2 ) dx + 2 xy dy = 0 . Example 4.2.6 Solve the initial value problem, xy dy = 2y 2 + 4 x 2 dx ; y 2) = 4 ( . Example 4.2.7 Solve 2 x 3 y dx + x 4 + y 4 ) dy = 0 ( . C. LINEAR EQUATIONS Standard equation: dy + P x )y = Q x ) ( ( dx Integrating factor: e∫ P x ) dx ( Procedure of solving linear 1st order DE: a) Put the equation into standard form (the coefficient of ( b) Identify P(x) and compute e ∫ P x ) dx . ( c) Multiply the standard equation by e ∫ P x ) dx . d) Integrate both sides of the equation and solve for y. 7 dy must be 1). dx
- 8. Example 4.2.8 Solve the linear equation 2 y ' − 4 y =16e x . Example 4.2.9 Solve x dy +y +4 = 0. dx Example 4.2.10 Solve the initial value problem, dy = x 2 − 3x 2 y dx ; y 1) = 2 . ( D. EXACT EQUATIONS Standard equations: Has the properties: M(x, y) dx + N(x, y) dy = 0 ∂ M ∂ N = ∂ y ∂ x Example 4.2.11 Show that the equation( x 2 + y 2 ) dx + 2 xy + cos y ) dy = 0 is exact and solve it. ( Example 4.2.12 Solve the equation x dy − y dx = xy 2 dx 4.3: APPLICATIONS OF FIRST ORDER DIFFERENTIAL EQUATIONS 4.3.1: APPLICATIONS OF SEPARABLE VARIABLES 8
- 9. dP = kP dx A. POPULATION GROWTH Example 4.3.1 If the population of a country doubles in 50 years, in how many years will it be triple under the assumption that the rate of increase is proportional to the number of inhabitants? Example 4.3.2 According to United Nations data, the world population at the beginning of 1975 was approximately 4 billion and growing at a rate of about 2% per year. Assuming an exponential growth model, estimate the world population at the beginning of the year 2010. B. RADIOACTIVE DECAY A radioactive substance decomposes at a rate proportional to its mass. This rate is called decay rate. The half-life of a substance is the amount of time for the substance to be half of its initial mass. Example 4.3.3 The half-life of carbon-14 is 5568 years. Show that the formula for the mass, M at time t is M = M0 e −0.0001245 t , where M0 is the initial mass. Example 4.3.4 The radioactive element, carbon-14 has a half-life of 5750 years. If 100 gm of this element are present initially, how much will be left after 1000 years? Example 4.3.5 9
- 10. A radioactive isotope has an initial mass of 100 mg, which two years later is 75 mg. Find the expression for the amount of the isotope remaining at any time. What is its half-life? C. NEWTON’S LAW OF COOLING Newton’s law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surrounding. If we let T(t) be the temperature of the object at time t, and Ts be the temperature of the surroundings, then dT = k T − Ts ) where k is a constant. ( dt Example 4.3.6 A bottle of soda pop at room temperature (72 oF), is placed in a refrigerator where the temperature is 44oF. After half an hour the soda pop has cooled to 61 oF. a) What is the temperature of the soda pop after another half hour? b) How long does it takes for the soda pop to cool to 50 oF? Example 4.3.7 A thermometer reading 100oF is placed in a pan of oil maintained at 10 oF. What is the temperature of the thermometer when t = 10 sec, if its temperature is 60 oF when t = 4 sec? 4.3.2: APPLICATIONS OF LINEAR DIFFERENTIAL EQUATIONS A. MIXTURE PROBLEM A typical mixing problem involves a tank of fixed capacity filled with a thoroughly mixed solution of some substance, such as salt. A solution of a given concentration enters the tank at a fixed rate and the mixture, thoroughly stirred, leaves at a fixed rate, which may differ from the entering rate. If y(t) denotes the amount of substance in the tank at time t, then y’(t) is the rate at which the substance is being added minus the rate at which it is being removed. 10
- 11. dy = (rate in) − (rate out ) dt Example 4.3.8 A tank contains 20 kg of salt dissolved in 5000 liter of water. Brine that contains 0.03 kg of salt per liter of water enters the tank at a rate of 25 l/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt remains in the tank after half an hour? Example 4.3.9 A tank is filled with 10 liter of brine in which is dissolved 5 gm of salt. Brine containing 3 gm of salt per liter enters the tank at 2 liter per min, and the well-stirred mixture leaves at the same rate. a) Find the amount of salt in the tank at any time. b) How much salt is present after 10 min? c) How much salt is present after a long time? Example 4.3.10 A tank with a capacity of 300 liters initially contains 100 liter of pure water. A salt solution containing 3 gm of salt per liter is allowed to run into the tank at a rate of 8 liter/min, and the mixture is then removed at a rate of 6 liter/min. Find the expression for number of gm of salt in the tank at any time, t. B. ELECTRIC CIRCUITS 1. 11
- 12. In a series circuit containing only a resistor, R and an inductor, L, Kirchoff’s Second Law states di and the voltage drop across the dt that the sum of the voltage drop across the inductor L resistor (iR) is the same as the impressed voltage (E(t)) on the circuit. Therefore: VL + VR = E(t) L di + Ri = E(t) dt di R E( t ) + i= ……………. Linear, 1st order, Differential Equation dt L L 2. The voltage drop across a capacitor with capacitance C, is given by on the capacitor. Hence, VR + VC = E(t) Ri + R 1 q = E(t) C dq 1 + q = E( t dt C where i = 12 dq dt q( t ) where q is the charge C
- 13. dq 1 + q = E( t ) ……………. Linear, 1st order, Differential Equation dt RC Example 4.3.11 A generator having emf 100 volts, is connected in series with a 10 ohms resistor and an inductor of 2 Henries. If the switch k is closed at time, t = 0, set up a differential equation for the current and determine the current at time t. Example 4.3.12 At t = 0, an emf of 100 sin 10t volts is applied to a circuit consisting of an inductor of 2 Henries in series with a 40 ohms resistor. If the current is zero at t = 0, what is it at anytime t ≥ 0? 4.4 SECOND ORDER DIFFERENTIAL EQUATIONS There are various types of second order differential equations but we will only consider the linear equation with constant coefficients. The general linear differential equation of the n th order is: a0 ( x) dn y dx n + a1( x ) dn −1y dx n −1 + ... + a n −1( x ) dy + a n ( x )y = f ( x ) dx , Where f(x) and the coefficients a i ( x )(i = 0, 1 2 ... n) depends on variable x. , A linear differential equation has a constant coefficients if all the coefficients a i ( x )(i = 0, 1 2 ... n) are constants. If f(x) = 0 , the equation is homogeneous linear equation, and if f(x) ≠ 0, the equation is nonhomogeneous linear equation. 13

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