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

e1s2kat26 2021-01-28 Answered

One solution of the differential equation y"4y=0 is y=e2xFind a second linearly independent solution using reduction of order.

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Expert Answer

krolaniaN
Answered 2021-01-29 Author has 86 answers
Given differential equation is y4y=0 (1)
Given that First solution is y1=e2x
let general solution y=vy1
substitute y1=e2x
y=ve2x
Differentiate y=ve2x with respect to x
y=ddx(ve2x)
=vddxe2x+e2xddx(v)
=v2e2x+e2xv
=2ve2x+e2xv
hence, y=2ve2x+e2xv,
Again differentiate y=2ve2x+e2xv with respect to x
y=ddx(2ve2x+e2xv)
=2ddx(ve2x)+ddx(e2xv)
=4ve2x+2e2xv+2e2xv+e2xv
=4ve2x+4e2xv+e2xv
Hence, y=4ve2x+4e2xv+e2xv
Now, substitute y=4ve2x+4e2xv+e2xv and y=ve2x in equation (1) and simlify it
4ve2x+4e2xv+e2xv4ve2x=0
4e2xv+e2xv=0 (2)
Let w=v
w=v
Substitute these value in equation (2)
4e2xw+e2xw=0
e2x(4w+w)=0 (Take common as e2x)
Divide both side be e2x and simplify it
e2x(4w+w)e2x=0e2x
4w+w=0
Substract 4w from both sides and simplify it
4w4w+w=4w
w=4w
dwdx=4w
dww=4dx
Now integrate it
dww=4dx+c
lnw=4x+c
w=e4x+c
w=e4xec
Further simplify it
w=c1e4x (Since

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