Solve the second order linear differential equation using method of undetermined coefficients 3y''+2y'-y=x^2+1

Yasmin 2020-12-07 Answered
Solve the second order linear differential equation using method of undetermined coefficients
3y+2yy=x2+1
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avortarF
Answered 2020-12-08 Author has 113 answers
To solve:
The second order linear differential equation, 3y+2yy=x2+1 using method of undetermined coefficients.
3y+2y"y=x2+1
Rewrite the avove equation as
3D2y+2Dyy=x2+1
(3D2+2D1)y=x2+1
The auxiliary equation is
3m2+2m1=0
3m2+3mm1=0
3m(m+1)1(m+1)=0
(3m1)(m+1)=0
3m1=0 or m+1=0
m=13 or m=1
The complimentary function is
C.F.=C1e13x+C2ex
To find the particular integral using the method of undetermined coefficients :
3y2yy=x2+1
The most general linear combination of the functions in the family is
yp=Ax2+Bx+c
dydx=2Ax+B
d2ydx2=2A
Plug dydx and d2dx2 in 3y+2yy=x2+1, we have
3(2A)+2(2Ax+B)(Ax2+Bx+C)=x2+1
6A+4Ax+2BAx2BxC=x2+1
Ax2+(4AB)x+(6A+2BC)=x2+1
Equating the coefficients of x2,we have
A=1
A=1
Equating the coefficients of x, we have
4AB=0
4(1)B=0
B=4
Equating the constant term, we have
6A+2BC=1
6(1)+2(4)C=1
68C=1
14C=1
C=141
C=15
Therefore, the particular solution is
P.I.=Ax2+Bx+C
=x24x15
=(x2+4x+15)
The general solution for the given differential equation is
y=C.F.+P.I.y=C1e13x+C2ex(x2+4x+15)
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