Solve the following initial value problem and present its particular solution: 2. y" + y' – 2y = 0, y(0) = -4, y'(0) = -5 %3D %3D 3. y" + y' + 0.25y = 0, y(0) = 3, y'(0) = -3.5 %3D
Solve the following initial value problem and present its particular solution: 2. y" + y' – 2y = 0, y(0) = -4, y'(0) = -5 %3D %3D 3. y" + y' + 0.25y = 0, y(0) = 3, y'(0) = -3.5 %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Answer #2 & #3
![(2²+a2 + b) = 0
The solution to the ODE depends on the nature of the
roots of the characteristic equation
EULER IDENTITIES
ea+ib
o FOR DIS TRcseR FUS
d=4=2 SHARE.
= e"(cos b + i sin b)
%3D
1
cos y =(e'y + e-iy)
sin y = (ety – e-ly)
Two Distinct Real Roots
1
Real Double Root
2i
1
y = cze21*+ c2xe?2*
5(e +e-Y)
cosh y =
2
1
Complex Conjugate Roots 11 = a + ib 12 = a – ib
sinh y =(ey – e-")
ey = cosh y + i sinh y
e-y = cosh y - i sinh y
y = eax(C1 cos bx + c2 sin bx)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5fb32ed0-76c2-497f-a229-b674ab9212e4%2F808d759c-57f9-49d1-93a2-97d6bfb99cf1%2Fmbc5sdu_processed.png&w=3840&q=75)
Transcribed Image Text:(2²+a2 + b) = 0
The solution to the ODE depends on the nature of the
roots of the characteristic equation
EULER IDENTITIES
ea+ib
o FOR DIS TRcseR FUS
d=4=2 SHARE.
= e"(cos b + i sin b)
%3D
1
cos y =(e'y + e-iy)
sin y = (ety – e-ly)
Two Distinct Real Roots
1
Real Double Root
2i
1
y = cze21*+ c2xe?2*
5(e +e-Y)
cosh y =
2
1
Complex Conjugate Roots 11 = a + ib 12 = a – ib
sinh y =(ey – e-")
ey = cosh y + i sinh y
e-y = cosh y - i sinh y
y = eax(C1 cos bx + c2 sin bx)
![Solve the following initial value problem and present its particular solution:
2. y" + y' – 2y = 0,
y(0) = -4,
y'(0) = -5
%3D
3. y" + y' + 0.25y = 0,
y(0) = 3,
y'(0) = -3.5
%3D
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5fb32ed0-76c2-497f-a229-b674ab9212e4%2F808d759c-57f9-49d1-93a2-97d6bfb99cf1%2Ffjrr8rg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Solve the following initial value problem and present its particular solution:
2. y" + y' – 2y = 0,
y(0) = -4,
y'(0) = -5
%3D
3. y" + y' + 0.25y = 0,
y(0) = 3,
y'(0) = -3.5
%3D
%3D
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