If I let w=y'

  • Thread starter Turion
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[tex]y'''-5y''+6y'=8+2sinx[/tex]

If I let w=y', would I be able to solve this differential equation? I'm currently stuck and I just want to know if making this substitution is why I am stuck.
 

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  • #2
SteamKing
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Have you followed the procedure for solving a linear ODE?

First, solve the homogeneous equation.
Second, find the particular solution.
Third, add the particular solution to the complementary solution (of the homogeneous equation).
 
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arildno
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[tex]y'''-5y''+6y'=8+2sinx[/tex]

If I let w=y', would I be able to solve this differential equation? I'm currently stuck and I just want to know if making this substitution is why I am stuck.
Yes, that works nicely.
If we look at the homogenous system, you get the characteristic polynomial, with e^rx as trial solution:
r^2-5r+6=0, giving r=3 and r=2 as possibles. Furthermore, you have w_p1=4/3, for the constant particular solution.
Now, in order to solve for the particular solution 2sinx, you generally will need w_p=Asin(x)+Bcos(x)

Inserting this gives you the two equations in A and B

sin(x): -A+5B+6A=2 goes to: 5A+5B=2
cos(x):-B-5A+6B=0 goes to: 5B-5A=0
Thus, A=B=1/5.
 

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