Find the solution of the following Differential Equations y" -

Linda Seales

Linda Seales

Answered question

2021-12-27

Find the solution of the following Differential Equations yy6y=0,y(0)=6,y(0)=13.

Answer & Explanation

Dabanka4v

Dabanka4v

Beginner2021-12-28Added 36 answers

Given: yy6y=0
with y(0)=6,y(0)=13
we put m for first derivative and m2 for second derivative then we solve it
(m2m6)y=0
Solving auxilliary equation
m2m6y=0
(m+2)(m3)=0
Solution is
y(x)=Ae2x+Be3x
With first initial condition y(0)=6
6=A+B(1)
y(x)=2Ae2x+3Be3x
With second initial condition y(0)=13
2A+3B=13 (2)
solving (1) and (2)
A=1,B=5
y=e2x+5e3x
Jim Hunt

Jim Hunt

Beginner2021-12-29Added 45 answers

yy6y=0
y=ekx
(ekx)(ekx)6ekx=0
k2ekx+kekx6ekx=0
ekx(k2+k6)=0
k2+k6=0
(k2)(k+3)=0
k1=2;y1=e2x
k2=3;y2=e3x
Y=C1e2x+C2e3x
{C1+C2=32C13C2=1
{C1=3C22C13C2=1
{C1=3C22(3C2)3C2=1
{C1=3C262C23C2=1
{C1=3C25C2=5
{C1=3C2C2=1
{C1=2C2=1
Y=2e2x+e3x
karton

karton

Expert2022-01-09Added 613 answers

yy6y=0 y(0)=6;y(0)=13L{y}=s2Y(s)sy(0)y(0)L{y}=sY(s)y(0)6L{y}=6Y(s)s2Y(s)sy(0)y(0)[sY(s)y(0)]6Y(s)=0s2Y(s)6s13sY(s)+66Y(s)=0Y(s)[s2s6]=6s+7Y(s)[(s+2)(s3)]=6s+7Y(s)=6s+7(s+2)(s3)As+2+Bs3=6s+7A(s3)+B(s+2)=6s+7Thus if you let s=2 then:A=1If you let s=3 then:B=5Y(s)=1s+2+5s3L{Y(s)}=L1{1s+2}+5L1{1s3}L{eet}=1say(t)=(1)e2t+5e3t

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