Solve the differential equation by variation of parameters. y''+3y'+2y=(1/9+e^x)

chanyingsauu7

chanyingsauu7

Answered question

2021-11-12

Solve the differential equation by variation of parameters.
y+3y+2y=(19+ex)

Answer & Explanation

Kara Dillon

Kara Dillon

Beginner2021-11-13Added 12 answers

To find solution of differential equation y+Py+Qy=R using variation of parameters, first we find complementary function and the using this we find complete solution.
Given that y+3y+2y=(19+ex)
This is second order differential equation with constant coefficient.
Here, R=(19+ex), P=3, Q=2
The auxiliary equation is
m2+3m+2=0
m2+2m+m+2=0
m(m+2)+1(m+2)=0
(m+2)(m+1)=0
m=1, 2
Complementary function is
yc=c1ex+c2e2x
Here, c1, c2 are arbitrary constant
Take, y1=ex, y2=e2x
w=[y1y2y1y2]=[exe2xex2e2x]=2e3x+e3x=e3x
Particular integral is
yp=Ay1+By2
Here, A=y2Rwdx
and B=y1Rwdx
A=e2xe3x(19+ex)dx
A=+ex(19+ex)dx
A=(ex9+e2x)dx
A=ex9+e2x2
B=y1Rwdx

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