3) ГУ" +у - бу — (12 — 16х)е*.iConsider the nonhomogeneous DE:A) Verify that y, = 4xe* is a particularsolution of the DE.B) Find the general solution of the given DE, ife2x and e-3xy" + y' – 6y = 0.are both solutions of

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Answered: 3)L. y" + y – 6y = (12 – 16x)e*. i…

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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B. Prove that the given equation is a solution to the given DE: 7. y = Ax + Be* ; (x- 1) y" - xy' + y = 0 8. y = Ce* + Cze2* + x ; y"- 3y' + 2y = 2x - 3
Q1.2 a find the biggest interval on which theorem 3.1.1 guarantees that the initial value problem has a unique 1 (x + 5) y" + (x³²-4) y² + 17y= 2² +1² 460) = 3₂ y ₁ (0) = 69 4" (0)=6 ' x +I³ options a) (-∞, -1) e) (-5,1) i) (-1,5) answer var. problem has b find the biggest interval on which rantees that the g unique (۱ - ره-)(e following solution. i)(-5, ∞0) b) (0,5) f) (-∞, -5) j) (-5, -1) 1) (1,5) Note: c) (-∞0, 1) g) (1, -∞) 1 (x - 57y" + (x²-4)y" + 17y = = = = = 54(0) = 3₂ y'(0) = 6+y" (01=6 + 1 Answer Options a) (-∞0,1) b) (1,00) f)(1,5) j) (-1,5) d) (5,00) 4) (5, ∞0) 4) (-1,00) theorem 3.1.1 following initial value solution. c)(5,1) 9) (-1,00) K) (-∞,5) d) (5,∞0) W)(-∞,-5) 2)(-5, -1) Theorem 3.1.1 Existence of a Unique Solution Let an(x), an-1(x), ..., a₁(x), a (x), and g(x) be continuous on an interval I, and let a,(x) = 0 for every x in this interval. If x= xo is any point in this interval, then a solution y(x) of the initial-value problem (1) exists on the interval and is…
Q13. If u= u(x,y) then the partial differential equation 2u, u, is of the kind a) Hyperbolic b) Elliptic c) Parabolic d) cyclic Q14. The solution of the differential equation y (1-x)+yx-y=0 is given by a) y = ze* b) y = 46²x c) y = 2x³ d) y = 2x³ -2-
2.5 (a) Prove that if x, y is a solution to ax+by = d, with d = gcd(a, b), then for all cЄZ, x = x + c b d' y = y_c. a (2.6.1) is also a solution to ax + by = d. (b) Find two distinct solutions to 2261x + 1275y = 17. (c) Prove that all solutions are of the form (2.6.1) for some c.
7:32 < Abdijabaar calculus 3.png Calculus 3 '1. Using z = y2, convert the following non-linear DE into linear DE. x2y dy dx - xy2 = 13; y(1) = 1 Solve the linear equation. Solutions: i) Differentiate z = y2 to get dy dx and replace into the non-linear equation. ii) Change the equation into the general form of linear equation & determine p(x) and q(x) iii) Find the integrating factor, µ(x) = eſp(x)dx iv) Find y v) Use the initial condition given, y(1) = 1.
1- If we use MUC to find y, of y"-5y'+4y= 4xe* + 2e* + 2.xcos3x. Then find the suitable form for 2- Find the largest possible interval in which the given DE has a unique solution tan(x)y" +xy' - y = xr' such that x, =-2,x, =5.
1.Find a general solution of each O.D.Es. A) y" 19y = coqx +os 3 - šin 2 (3) y" + 4y/ + 6.25у — 3.125(х +1)2. (2)y" 2x 3.125(x + 1)2.
Show that x(t) = aje+njt + aze- , jt.is equal to Asin(@n+P). %3D 3. Using the general solution: x(t) = A1cos(@nt)+A2sin(@nt) siculate the values of A1 , A2 in terms of their initial conditions x0, v0
) Find the General solution of y" - 4y' + 4y = 9te
Exercise 7.3.5: Find a particular (Frobenius-type) solution of y"+y' - xy = 0. A conseios com Exercise 7.3.6: Find the general solution of 2xy"+y' - x²y = 0. Exercise 7.3.7: Find the general solution of x²y" - xy' - y = 0. stout board
The general solution of the nonhomogeneous equation y "-y = x is a. y = C + C2 ex – x + 1 b. y = C cOS x + C2 sin x + x C. y= C ex + C2 e* - x d. y = c, cos x + C2 sin x + x + 1

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3)L. y" + y – 6y = (12 – 16x)e*. i Consider the nonhomogeneous DE: A) Verify that y, = 4xe* is a particular solution of the DE. B) Find the general solution of the given DE, if e2x and e-3x are both solutions of y" + y' – 6y = 0.

3)L. y" + y – 6y = (12 – 16x)e. i Consider the nonhomogeneous DE: A) Verify that y, = 4xe is a particular solution of the DE. B) Find the general solution of the given DE, if e2x and e-3x are both solutions of y" + y' – 6y = 0.