Homoge 5. The equation a?y" + axy + by = 0 where a.b are constants is called Euler-Cauchy equation. Show that under the transfor- mation r = e' when r >0 the above reduces to y" + (a – 1)/ + by = 0 which is an equation with constant coefficients. Hence solve (i) ry" + 2ary – 12y = 0 (ii) a?y" + xy + y = 0 (iii) a?y" – xy + y = 0 6. Find differential equation with constant coefficients satisfied by the functions (ii) e-2" cos 3r, e-2r sin 3r, e-2x re-27. (i) a2, e", re" 7. Find the general solution of the following differential equations. d'y (a) + y(x) = 0. dri d³y (b) dr d'y dy = 0. dr dz4 By dy dy (c) dr ই + dr - y(x) = 0. d°y (d) d'y + 10 dr dy Ts + 10 +5 fip² °dz dr² + y(x) = 0. 8. Solve y" +y = tana in the interval (0, 7) using the method of variation of parameter. Answer: In | sect| +1- sin t(ln | sec t + tan t|). 9. Solve the following initial-value problems: (a) y" – 2y + y = 2re2" + 6e"; y(0) = 1, y'(0) = 0. (b) y"(x) + y(x) = 3x2 – 4 sin r, y(0) = 0, y'(0) = 1. 10. Use the method of undermined coefficients to find a particular solution to the following differential equations: (a) y" – 3y' + 2y = 2x² + 3e2r. (b) y"(r) – 3y (r)+ 2y(x) = xe²# + sin æ.
Homoge 5. The equation a?y" + axy + by = 0 where a.b are constants is called Euler-Cauchy equation. Show that under the transfor- mation r = e' when r >0 the above reduces to y" + (a – 1)/ + by = 0 which is an equation with constant coefficients. Hence solve (i) ry" + 2ary – 12y = 0 (ii) a?y" + xy + y = 0 (iii) a?y" – xy + y = 0 6. Find differential equation with constant coefficients satisfied by the functions (ii) e-2" cos 3r, e-2r sin 3r, e-2x re-27. (i) a2, e", re" 7. Find the general solution of the following differential equations. d'y (a) + y(x) = 0. dri d³y (b) dr d'y dy = 0. dr dz4 By dy dy (c) dr ই + dr - y(x) = 0. d°y (d) d'y + 10 dr dy Ts + 10 +5 fip² °dz dr² + y(x) = 0. 8. Solve y" +y = tana in the interval (0, 7) using the method of variation of parameter. Answer: In | sect| +1- sin t(ln | sec t + tan t|). 9. Solve the following initial-value problems: (a) y" – 2y + y = 2re2" + 6e"; y(0) = 1, y'(0) = 0. (b) y"(x) + y(x) = 3x2 – 4 sin r, y(0) = 0, y'(0) = 1. 10. Use the method of undermined coefficients to find a particular solution to the following differential equations: (a) y" – 3y' + 2y = 2x² + 3e2r. (b) y"(r) – 3y (r)+ 2y(x) = xe²# + sin æ.
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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Transcribed Image Text:Homoge
5. The equation
a?y" + axy + by = 0
where a.b are constants is called Euler-Cauchy equation. Show that under the transfor-
mation r = e' when r >0 the above reduces to
y" + (a – 1)/ + by = 0
which is an equation with constant coefficients. Hence solve
(i) ry" + 2ary – 12y = 0 (ii) a?y" + xy + y = 0
(iii) a?y" – xy + y = 0
6. Find differential equation with constant coefficients satisfied by the functions
(ii) e-2" cos 3r, e-2r sin 3r, e-2x re-27.
(i) a2, e", re"
7. Find the general solution of the following differential equations.
d'y
(a)
+ y(x) = 0.
dri
d³y
(b)
dr
d'y dy
= 0.
dr
dz4
By dy
dy
(c)
dr
ই +
dr
- y(x) = 0.
d°y
(d)
d'y
+ 10
dr
dy
Ts + 10 +5
fip²
°dz
dr²
+ y(x) = 0.
8. Solve y" +y = tana in the interval (0, 7) using the method of variation of parameter.
Answer: In | sect| +1- sin t(ln | sec t + tan t|).
9. Solve the following initial-value problems:
(a) y" – 2y + y = 2re2" + 6e"; y(0) = 1, y'(0) = 0.
(b) y"(x) + y(x) = 3x2 – 4 sin r, y(0) = 0, y'(0) = 1.
10. Use the method of undermined coefficients to find a particular solution to the following
differential equations:
(a) y" – 3y' + 2y = 2x² + 3e2r.
(b) y"(r) – 3y (r)+ 2y(x) = xe²# + sin æ.
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