One solution of the differential equation y" – 4y = 0 is y = e^{2x} Find a second linearly independent solution using reduction of order.

e1s2kat26

e1s2kat26

Answered question

2021-01-28

One solution of the differential equation y"4y=0 is y=e2x. Find a second linearly independent solution using reduction of order.

Answer & Explanation

krolaniaN

krolaniaN

Skilled2021-01-29Added 86 answers

General solution y=vy1
y=ve2x
Differentiate y=ve2x with respect to x
y=ddx(ve2x)=vddxe2x+e2xddx(v)=v2e2x+e2xv=2ve2x+e2xv
Differentiate y=2ve2x+e2xv with respect to x
y=ddx(2ve2x+e2xv)=2ddx(ve2x)+ddx(e2xv)=4ve2x+2e2xv+2e2xv+e2xv
=4ve2x+4e2xv+e2xv
4ve2x+4e2xv+e2xv4ve2x=0
4e2xv+e2xv=0 (2)
Let w=v
w=v
4e2xw+e2xw=0
e2x(4w+w)=0, take common as e2x
e2x(4w+w)e2x=0e2x
4w+w=0
4w4w+w=4w
w=4w
dwdx=4w
dww=4dx
dww=4dx+c
lnw=4x+c
w=e4x+c
w=e4xec
w=c1e4x 

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