(10) Use the result of Exercise 9 to find the general solution of the given equation: (a) (r-3)2y"+3(r -3)y+y = 0, r > 3 (b) (2r +1)2/" + 4(2.r + 1)y' – 24y = 0, r> - 1

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Exercises. Solve the given equation, if there is an initial value, find the solution of the IVP. Assume that r>0 in all
of the problems.
(1) y"- 2y = 0
(2) a2y" + 4xy + 2y = 0; y(1) = 1, y' (1) = 2
(3) r?y" + 3ry' -3y = 0
(4) ay"-3.ry + 4y = 0; y(1) = 2, y/ (1) = 1
(5) 3.ry" + 2y' = 0
(6) 4.r2y" + y = 0
(7) y"+ xy + 4y = 0; y(1) = 1, y (1) = 4
(8) a?y" – 5xy + 13y = 0
Challenge Problems:
(9) Show that the change of variable t = ar +b transforms the equation
bo(a.r +b)y" + by (ar +b)y + b2y = 0
into a Cauchy-Euler equation.
(10) Use the result of Exercise 9 to find the general solution of the given equation:
(a) (r-3)2y" +3(x - 3)y +y = 0, r > 3
(b) (2r + 1)2y" + 4(2.x + 1)y' – 24y = 0, r > -
1
Transcribed Image Text:Exercises. Solve the given equation, if there is an initial value, find the solution of the IVP. Assume that r>0 in all of the problems. (1) y"- 2y = 0 (2) a2y" + 4xy + 2y = 0; y(1) = 1, y' (1) = 2 (3) r?y" + 3ry' -3y = 0 (4) ay"-3.ry + 4y = 0; y(1) = 2, y/ (1) = 1 (5) 3.ry" + 2y' = 0 (6) 4.r2y" + y = 0 (7) y"+ xy + 4y = 0; y(1) = 1, y (1) = 4 (8) a?y" – 5xy + 13y = 0 Challenge Problems: (9) Show that the change of variable t = ar +b transforms the equation bo(a.r +b)y" + by (ar +b)y + b2y = 0 into a Cauchy-Euler equation. (10) Use the result of Exercise 9 to find the general solution of the given equation: (a) (r-3)2y" +3(x - 3)y +y = 0, r > 3 (b) (2r + 1)2y" + 4(2.x + 1)y' – 24y = 0, r > - 1
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