Give the correct answer and solve the given equation: displaystyle{y} text{ - 4y}+{3}{y}={x},{y}_{{1}}={e}^{x}

he298c 2021-02-25 Answered
Give the correct answer and solve the given equation: y  - 4y+3y=x,y1=ex
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Expert Answer

Sadie Eaton
Answered 2021-02-26 Author has 104 answers

The homogeneous equation is y4y+3y=0
Its charecteristic equation is
r24r+3=0r=4±(4)241321=4±22=2±1
Thus, the solutions of the characteristic equation are r1=3,r2=1, so the solution of (1) is
y=C1er1x+C2er2x=C1e3x+C2ex
where C1andC2 are constants.
Now we need to find a particular solution, that is, we have to guess some function y such that y4y+3y=x(2)
Since the RHS is a polynomial of degree 1, we will try with y=ax+b.
Then y=a,y=0
so (2) becomes
4a+3ax+3x=x3ax+(3ba)=1x
From this we get
3a=1
3b4a=0
Solving this system we get
a=13,3b=4a=43b=49
Therefore, the particular solution is
y=ax+b=13x+49
Finally, the solution of the original equation is given by the sum of the homogeneous and particular solution, so
y=C1e3x+C2ex+13x+49

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