Consider the following linear second order homogeneous differential

ideklaraz7xz

ideklaraz7xz

Answered question

2022-03-24

Consider the following linear second order homogeneous differential equation wit constant coefficients and two initial conditions
d2y(t)dt2+dy(t)dt2y(t)=0, y(0)=1 dy(0)dt=5
Solve the differential equation.

Answer & Explanation

haiguetenteme7zyu

haiguetenteme7zyu

Beginner2022-03-25Added 13 answers

Solution:
Given:d2y(t)dt2+dy(t)dt2y(t)=0, (1)
y(0)=1 and y(0)=5
Auxiliary equation is m2+m2=0
m2+2mm2=0
m(m+2)1(m+2)=0
(m+2)(m1)=0
m1=2 and m2=1 are the roots of auxiliary equation. Then general solution will be
y(t)=c1e2t+c2et (2)
where c1 and c2 are constant derivative of (2) with respect to t, we get
ddty(t)=ddt[c1e2t+c2et]
y(t)=c1ddte2t+c2ddtet
y(t)=2c1e2t+c2et (3)
y(0)=1 then from (2)
c1e0+c2e0=1
c1+c2=1 (4)
Also y(0)=5. Then from (3)
2c1e0+c2e0=5
2c1+c2=5 (5)
Solving equation (4) and (5) we get
c2=1 and c1=2
using these value of c1 and c2 in equation (2) we get
y(t)=2e2t+et
Jeffrey Jordon

Jeffrey Jordon

Expert2022-03-31Added 2605 answers

Answer is given below (on video)

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