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Differential equations Assignment 13 This assignment covers the material contained in Chapters 46 to 49. The marks for each question are shown in brackets at the end of each question. 1. Solve the differential equation: x dy dx + x2 = 5 given that y = 2.5 when x = 1. (4) 2. Determine the equation of the curve which satisfies the differential equation 2xy dy dx = x2 + 1 and which passes through the point (1, 2). (5) 3. A capacitor C is charged by applying a steady voltage E through a resistance R. The p.d. between the plates, V, is given by the differential equation: CR dV dt + V = E (a) Solve the equation for E given that when time t = 0, V = 0. (b) Evaluate voltageV when E =50 V, C =10 µF, R = 200 k and t = 1.2 s. (14) 4. Show that the solution to the differential equation: 4x dy dx = x2 + y 2 is of the form y 3y 2 = √ x (1 − √ ) x 3 given that y = 0 when x = 1 (12) 5. Show that the solution to the differential equation x cos x dy + (x sin x + cos x)y = 1 dx is given by: xy = sin x + k cos x where k is a constant. (11) 6. (a) Use Euler’s method to obtain a numerical solution of the differential equation: dy dx = y x + x2 − 2 given the initial conditions that x = 1 when y = 3, for the range x = 1.0 (0.1) 1.5. (b) Apply the Euler-Cauchy method to the differential equation given in part (a) over the same range. (c) Apply the integrating factor method to solve the differential equation in part (a) analytically. (d) Determine the percentage error, correct to 3 significant figures, in each of the two numerical methods when x = 1.2. (30) 7. Use the Runge-Kutta method to solve the differential equation: dy dx = y x + x2 − 2 in the range 1.0(0.1)1.5, given the initial conditions that at x = 1, y = 3. Work to an accuracy of 6 decimal places. (24)
Differential equations 50 Second order differential equations of the form a d2 y dx 2 + bdy dx + cy = 0 50.1 Introduction An equation of the form a d2 y dy +b + cy = 0, where dx2 dx a, b and c are constants, is called a linear second order differential equation with constant coefficients. When the right-hand side of the differential equation is zero, it is referred to as a homogeneous differential equation. When the right-hand side is not equal to zero (as in Chapter 51) it is referred to as a non-homogeneous differential equation. There are numerous engineering examples of second order differential equations. Three examples are: (i) L d2 q dt 2 + Rdq dt + 1 q = 0, representing an equation for charge q in an electrical circuit contain- C ing resistance R, inductance L and capacitance C in series. (ii) m d2 s dt 2 + ads + ks = 0, defining a mechanical dt system, where s is the distance from a fixed point after t seconds, m is a mass, a the damping factor and k the spring stiffness. (iii) d2 y dx 2 + P y = 0, representing an equation for EI the deflected profile y of a pin-ended uniform strut of length l subjected to a load P. E is Young’s modulus and I is the second moment of area. If D represents d dx and D2 represents d2 then the dx2 above equation may be stated as (aD 2 + bD + c)y = 0. This equation is said to be in ‘D-operator’ form. If y = Ae mx then dy dx = Amemx and d2 y dx 2 = Am2 e mx . Substituting these values into a d2 y dx 2 + b dy dx + cy = 0 gives: a(Am 2 e mx ) + b(Ame mx ) + c(Ae mx ) = 0 i.e. Ae mx (am 2 + bm + c) = 0 Thus y = Ae mx is a solution of the given equation provided that (am 2 + bm + c) = 0. am 2 + bm + c = 0is called the auxiliary equation, and since the equation is a quadratic, m may be obtained either by factorising or by using the quadratic formula. Since, in the auxiliary equation, a, b and c are real values, then the equation may have either (i) two different real roots (when b 2 > 4ac) or (ii) two equal real roots (when b 2 = 4ac) or (iii) two complex roots (when b 2 < 4ac). 50.2 Procedure to solve differential equations of the form a d2 y dx 2 + bdy dx + cy = 0 (a) Rewrite the differential equation a d2 y dx 2 + b dy dx + cy = 0 as (aD 2 + bD + c)y = 0 (b) Substitute m for D and solve the auxiliary equation am 2 + bm + c = 0 for m. I
Page 1 and 2: Differential equations 46 Solution
Page 3 and 4: SOLUTION OF FIRST ORDER DIFFERENTIA
Page 5 and 6: α is the temperature coefficient o
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Page 11 and 12: HOMOGENEOUS FIRST ORDER DIFFERENTIA
Page 13 and 14: Differential equations 48 Linear fi
Page 15 and 16: Hence from equation (4): ye x = xe
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Page 19 and 20: NUMERICAL METHODS FOR FIRST ORDER D
Page 31: NUMERICAL METHODS FOR FIRST ORDER D
Page 35 and 36: SECOND ORDER DIFFERENTIAL EQUATIONS
Page 37 and 38: When thus t = 0, dV dt = 3ω, 3ω =
Page 39 and 40: Differential equations 51 Second or
Page 49 and 50: Differential equations 52 Power ser
Page 51 and 52: POWER SERIES METHODS OF SOLVING ORD
Page 71 and 72: AN INTRODUCTION TO PARTIAL DIFFEREN
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Differential equations Assignment 1
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