Consider the following linear second-order homogeneous differential equation

Coradossi7xod 2022-03-25 Answered
Consider the following linear second-order homogeneous differential equation with constant coefficients and two initial conditions
d2y(t)dt213dy(t)dt29y(t)=0, y(0)=0, dy(0)dt=3
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Answers (2)

Korbin Ochoa
Answered 2022-03-26 Author has 11 answers
Solution:
d2y(t)dt213dy(t)dt29y(t)=0
y(0)=0, dy(0)dt=3
Auxiliary equation:
m213m29=0
9m23m2=0
(m23)(m+13)=0 (factor form)
m=23, 13
General solution is
y(t)=c1e23t+c2e13t (1)
Apply intral conditions
y(0)=0
0=c1e0+c2e0c1+c2=0 (2)
and from (1)
dydt(t)=23c1e23t13c2e13t
Apply dy(0)dt=33=23c1e013c2e09=2c1c2 (3)
and equation (2) and (3)
2c1c2+c1+c2=9
3c1=9c1=3
By equation (2) c2=3
Plug value of c1=3 and c2=3 in equation (1) we get
y(t)=3e23t+3e13t

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Jeffrey Jordon
Answered 2022-03-31 Author has 2495 answers

Answer is given below (on video)

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