Re: [考古] 99成大微分方程
※ 引述《dali510313 (乳)》之銘言：
: For a differential equation x^2 y" -3xy' +4y = 0,
: (a)
: use z = lnx to transform such an equation into an equation with constant
: coefficients
: (b)
: find the general solution of (a) in terms of x.
: 不好意思,又來打擾各位了。這題讓我想了好久，是假設ｙ的積次因子嗎？
: 看不太懂題意。先謝謝各位了。
原本題目應該是x平方*y''
we don't know the relation between y and x ..
dz 1 dx dy
but from z = lnx we know ---- = ---- and it can be ---- ---- = y'x = k'
dx x dz dx
k' means y differential respect to z
chain rule
2
dy' d dy dz dz d y dz dy -1
--- = ---- (---- ----) = --- ---- *---- + ---- ----
2 2
dx dx dz dx dx dz dx dz x
from above we know dz/dx = 1/x ,
so we know y'' = (k'' - k ')/x^2
and y' = k' /x
put they into the original equation. x^2 y'' - 3 x y' + 4 y = 0
we get k'' - k' - 3k' + 4y = 0
k'' - 4k' + 4 = 0
get new equation with new constant coefficents.
tz
slove it , because constant coefficents , use e = y
put the solution into the equation.
2
t - 4 t + 4 = 0 t = 2.
2z 2z
so the basis of solution is e and ze
z
and z = ln x so , x = e
The general solution is y = c1(x^2) + c2 (lnx*x^2)
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07/06 17:52, , 1F
07/06 17:52, 1F
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