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artcher [175]
3 years ago
5

Find a particular solution to y′′+25y=−40sin(5t). y″+25y=−40sin⁡(5t).

Mathematics
1 answer:
daser333 [38]3 years ago
4 0

Answer:

Step-by-step explanation:

Non homogeneous differential

y''+25y = -40sin(5t)---(1)

To find homogeneous solution consider auxilary equation:

D^{2}+25=0\\D=\pm5i

Homogeneous solution is:

y_{h}=Acos(5t)+Bsin(5t)\\

Particular soution for -40 sin(5t)

y_{p}= Csin(5t)+Dcos(5t)\\y'_{p}=5Ccos(5t)-5Dsin(5t)\\y''_{p}=-25Csin(5t)-25Dcos(5t)\\

substituting value of y_{p}, y''_{p}  in (1)

-25Csin(5t)-25Dcos(5t)+25Csin(5t)+25Dcos(5t)=-40sin(5t)\\\implies\\C=0\\D=0\\\implies \\y_{p}=0

General solution is

y=Acos(5t)+Bsin(5t)

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