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Solve quetion 3.5.7.9

Solve quetion 3.5.7.9

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve quetion 3.5.7.9

dx = x + y? dy dx - x = y dy or is recognized as linear in the variable x. You should verify that the integrating factor e-1)dy = e¯y and integration by parts yield the explicit solution x = -y² – 2y - 2 + ce" for the second equation. This expression is, then, an implicit solution of the first equation. (iii) Mathematicians have adopted as their own certain words from engineer- ing, which they found appropriately descriptive. The word transient, used earlier, is one of these terms. In future discussions the words input and output will occasionally pop up. The function f in (2) is called the input or driving function; a solution y(x) of the differential equation for a given input is called the output or response. (iv) The term special functions mentioned in conjunction with the error func- tion also applies to the sine integral function and the Fresnel sine integral introduced in Problems 49 and 50 in Exercises 2.3. "Special Functions" is actually a well-defined branch of mathematics. More special functions are studied in Section 6.3. *Certain commands have the same spelling, but in Mathematica commands begin with a capital letter (Dsolve), whereas in Maple the same command begins with a lower case letter (dsolve). When discussing such common syntax, we compromise and write, for example, dsolve. See the Student Resource and Solutions Manual for the complete input commands used to solve a linear first-order DE. EXERCISES 2.3 Answers to selected odd-numbered problems begin on page ANS-2. In Problems 1-24 find the general solution of the given dif- ferential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. 5. y' + 3x²y = x² 6. y' + 2xy = x³ 7. х?у + ху 3 1 8. y' = 2y + x² + 5 dy dy 9. х y = x² sin x 10. х + 2y = 3 dy dy dx dx 1. = 5y 2. + 2y = 0 dx dx dy dy 12. (1 + x) – xy = x + x² dx 11. x + 4y = x³ – x dy dy ´dx 3. + y = e3r 4. 3 + 12y = 4 dx dx 13. x?y' + x(x + 2)y = e* 2.3 LINEAR EQUATIONS 61 dy 14. xy' + (1 + x)y = e¬* sin 2x 33. + 2xy = f(x), y(0) = 2, where dx 15. у dx — 4(х + yб) dy 3D 0 (x, 0<x<1 f(x) = [0, 16. у dx %3D (yеУ — 2х) dy x21 dy + (sin x)y = 1 dx 17. cos x dy + 2xy = f(x), y(0) = 0, where dx 34. (1 + x?) dy + (cos'x)y = 1 dx 18. cos?x sin x Sx, f(x) = 0 <x<1 dy + (x + 2)y = 2xe¯* dx 19. (х + 1) — 35. Proceed in a manner analogous to Example 6 to solve the initial-value problem y' + P(x)y = 4x, y(0) = 3, where dy 3 5 — 8у — 4ху dx 20. (x + 2) 2, P(x) = 0 <x<1, —2/х, х> 1. dr 21. + r sec 0 = cos 0 de Use a graphing utility to graph the continuous function y(x). dP 22. + 2tP = P + 4t – 2 dt 36. Consider the initial-value problem y' + e*y = f(x), y(0) = 1. Express the solution of the IVP for x>0 as a nonelementary integral when f(x) = 1. What is the so- lution when f(x) = 0? When f(x) = e*? dy 23. х + (3x + 1)y = e¯3x dx dy 24. (x² – 1) + 2y = (x + 1)² dx 37. Express the solution of the initial-value problem y' – 2xy = 1, y(1) = 1, in terms of erf(x). In Problems 25– 30 solve the given initial-value problem. Give the largest interval I over which the solution is defined. 25. ху' + у %3D e', у(1) %3 2 Discussion Problems dx 38. Reread the discussion following Example 2. Construct a linear first-order differential equation for which all nonconstant solutions approach the horizontal asymp- tote y = 4 as x→ 0. 26. У - x = 2y², y(1) = 5 dy di + Ri = E, i(0) = io, dt 27. L 39. Reread Example 3 and then discuss, with reference to Theorem 1.2.1, the existence and uniqueness of a solution of the initial-value problem consisting of xy' – 4y = x°e* and the given initial condition. (b) у(0) 3D yо, yo > 0 L, R, E, and io constants LP = k(T – T„); T(0) = T0, dt 28. k, Tm, and To constants (а) у(0) — 0 dy (c) y(xo) = yo, xo > 0, yo > 0 + y = In x, y(1) = 10 dx 29. (х + 1) 40. Reread Example 4 and then find the general solution of the differential equation on the interval (–3, 3). 30. y' + (tan x)y = cos²x, y(0) = -1
Transcribed Image Text:dx = x + y? dy dx - x = y dy or is recognized as linear in the variable x. You should verify that the integrating factor e-1)dy = e¯y and integration by parts yield the explicit solution x = -y² – 2y - 2 + ce" for the second equation. This expression is, then, an implicit solution of the first equation. (iii) Mathematicians have adopted as their own certain words from engineer- ing, which they found appropriately descriptive. The word transient, used earlier, is one of these terms. In future discussions the words input and output will occasionally pop up. The function f in (2) is called the input or driving function; a solution y(x) of the differential equation for a given input is called the output or response. (iv) The term special functions mentioned in conjunction with the error func- tion also applies to the sine integral function and the Fresnel sine integral introduced in Problems 49 and 50 in Exercises 2.3. "Special Functions" is actually a well-defined branch of mathematics. More special functions are studied in Section 6.3. *Certain commands have the same spelling, but in Mathematica commands begin with a capital letter (Dsolve), whereas in Maple the same command begins with a lower case letter (dsolve). When discussing such common syntax, we compromise and write, for example, dsolve. See the Student Resource and Solutions Manual for the complete input commands used to solve a linear first-order DE. EXERCISES 2.3 Answers to selected odd-numbered problems begin on page ANS-2. In Problems 1-24 find the general solution of the given dif- ferential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. 5. y' + 3x²y = x² 6. y' + 2xy = x³ 7. х?у + ху 3 1 8. y' = 2y + x² + 5 dy dy 9. х y = x² sin x 10. х + 2y = 3 dy dy dx dx 1. = 5y 2. + 2y = 0 dx dx dy dy 12. (1 + x) – xy = x + x² dx 11. x + 4y = x³ – x dy dy ´dx 3. + y = e3r 4. 3 + 12y = 4 dx dx 13. x?y' + x(x + 2)y = e* 2.3 LINEAR EQUATIONS 61 dy 14. xy' + (1 + x)y = e¬* sin 2x 33. + 2xy = f(x), y(0) = 2, where dx 15. у dx — 4(х + yб) dy 3D 0 (x, 0<x<1 f(x) = [0, 16. у dx %3D (yеУ — 2х) dy x21 dy + (sin x)y = 1 dx 17. cos x dy + 2xy = f(x), y(0) = 0, where dx 34. (1 + x?) dy + (cos'x)y = 1 dx 18. cos?x sin x Sx, f(x) = 0 <x<1 dy + (x + 2)y = 2xe¯* dx 19. (х + 1) — 35. Proceed in a manner analogous to Example 6 to solve the initial-value problem y' + P(x)y = 4x, y(0) = 3, where dy 3 5 — 8у — 4ху dx 20. (x + 2) 2, P(x) = 0 <x<1, —2/х, х> 1. dr 21. + r sec 0 = cos 0 de Use a graphing utility to graph the continuous function y(x). dP 22. + 2tP = P + 4t – 2 dt 36. Consider the initial-value problem y' + e*y = f(x), y(0) = 1. Express the solution of the IVP for x>0 as a nonelementary integral when f(x) = 1. What is the so- lution when f(x) = 0? When f(x) = e*? dy 23. х + (3x + 1)y = e¯3x dx dy 24. (x² – 1) + 2y = (x + 1)² dx 37. Express the solution of the initial-value problem y' – 2xy = 1, y(1) = 1, in terms of erf(x). In Problems 25– 30 solve the given initial-value problem. Give the largest interval I over which the solution is defined. 25. ху' + у %3D e', у(1) %3 2 Discussion Problems dx 38. Reread the discussion following Example 2. Construct a linear first-order differential equation for which all nonconstant solutions approach the horizontal asymp- tote y = 4 as x→ 0. 26. У - x = 2y², y(1) = 5 dy di + Ri = E, i(0) = io, dt 27. L 39. Reread Example 3 and then discuss, with reference to Theorem 1.2.1, the existence and uniqueness of a solution of the initial-value problem consisting of xy' – 4y = x°e* and the given initial condition. (b) у(0) 3D yо, yo > 0 L, R, E, and io constants LP = k(T – T„); T(0) = T0, dt 28. k, Tm, and To constants (а) у(0) — 0 dy (c) y(xo) = yo, xo > 0, yo > 0 + y = In x, y(1) = 10 dx 29. (х + 1) 40. Reread Example 4 and then find the general solution of the differential equation on the interval (–3, 3). 30. y' + (tan x)y = cos²x, y(0) = -1
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