−y′′−2y′+2y=e−2x
y(0)=1
y(∞)=0
Solution:
y′′+2y′−2y=−e−2x
Characteristic equation:
r2+2r−2=0 ;
D=4+4⋅2=23 ;
r1=−1+3 ; r2=−1−3 .
The general solution of the homogeneous differential equation:
y0=C1e(−1+3)x+C2e(−1−3)x
Let's find a partial solution of the non-homogeneous equation in the form
yp=Ae−2x
yp′=−2Ae−2x
yp′′=4Ae−2x
Substitute yp, yp′, yp′′ into the origin differential equation:
4Ae−2x−4Ae−2x−2Ae−2x=−e−2x
A=21
The general solution of the non-homogeneous differential equation:
y=y0+yp=C1e(−1+3)x+C2e(−1−3)x+21e−2x
Let's apply conditions:
y(0)=C1+C2+21=1
y(∞)=0⟹ C1=0 (because if C1=0 then y(∞)→∞ )
C2=21
Answer: y=21e(−1−3)x+21e−2x .
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