Solve this differential equation: NSK PSKy''-7y'+10y=\cos xZSK NSK where y is a one-variable function

interdicoxd 2021-12-07 Answered
Solve this differential equation:
y7y+10y=cosx
where y is a one-variable function x. What am I supposed to do with the cosx?
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Expert Answer

Ethan Sanders
Answered 2021-12-08 Author has 35 answers
You can solve by means of complex numbers, generalizing to
As you know that the derivatives of the exponential are the same exponential times a coefficient, the solution must be of the form ceix. Then
(i27i+10)ceix=eix
and c=197i=9+7i130
Finally, the solution is the real part of ceix, or
9cosx7sinx130
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redhotdevil13l3
Answered 2021-12-09 Author has 30 answers
For the homogeneous part, I am sure that you do not face any problem.
Now, suppose that the solution is
y=Asin(x)+Bcos(x)
y=Acos(x)Bsin(x)
y=Asin(x)Bcos(x)
and replace to get
(9A+7B)sin(x)+(9B7A)cos(x)=cos(x)
Identify; this gives
9A+7B=0
9B7A=1
Solve for A,B.
Just to make the problem more general, consider the equation
y+ay+by=csin(x)+dcos(x)
and just do the same as above. You will end with
aB+A(b1)c=0
aA+(b1)Bd=0
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