Solve the differential equation using either the method of undetermined coefficients or Laplace transforms. y'''-6y"+11y'-6y=3x

he298c

he298c

Answered question

2021-09-03

Solve the differential equation using either the method of undetermined coefficients or Laplace transforms.
y6ytext11y6y=3x

Answer & Explanation

Mayme

Mayme

Skilled2021-09-04Added 103 answers

Step 1
The differential equation is y6y11y6y=3x
Obtain the auxiliary equation as follows.
m36m2+11m6=0
(m1)(m2)(m3)=0
m=1,2and3
The roots are m1=1,m2=2  and  m3=3
Thus , the complimentary solution of the homogeneous equation is
yc(x)=c1ex+c2e2x+c3e3x
Step 2
Obtain the particular solution as follows.
Let yp(x)=Ax+B
Substitute yp(x)=Ax+B  in  y6y11y6=3x
(0)6(0)+11(A)6(Ax+B)=3x
11A6Ax6B=3x
6Ax=3x,11A6B=0 (By comparing like terms)
A=36,B=1112
Thus, the particular solution is yp(x)=12x1112
Therefore , the general solution is
y=c1ex+c2e2x+c3e3x12x1112
Step 3
Answer
The general solution of the differential equation is y=c1ex+c2e2x+c3e3x12x1112

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