Re: [微積] 兩題工數
※ 引述《Dooo (臘腸狗切半去尾)》之銘言：
: 章節:Exact ODEs. Integrating Factors
: 題目:2cosh(x)cos(y)dx=sinh(x)sin(y)dy
: Integrating Factor就算找到了我還是解不出來
: R(y)= (1/2)tan(y) ， F=(cosy)^(-1/2)
: 不知道有沒有求錯...
: 再追問一題好了
: Extended method
: xy'=y+3(x^4)(cos(y/x))^2 ，u=y/x ，y(1)=0
: 我算到最後會出現ln(-1)...
2cosh(x)cos(y)dx=sinh(x)sin(y)dy
2cos(y)d[sinh(x)]+sinh(x)d[cos(y)]=0
d[sinh^2(x)*cos(y)]
────────── =0
sinh(x)
1
同乘積分因子I= ───────
sinh(x)cos(y)
d[sinh^2(x)*cos(y)]
────────── =0
sinh^2(x)cos(y)
ln│sinh^2(x)cos(y)│=c1 為隱函數解 兩端開指數可得顯函數解
(2)
xy'=y+3(x^4)(cos(y/x))^2
xdu
let u=y/x , y=ux , y'= ───+u =xu'+u
dx
x(xu'+u) =ux+3(x^4)[cos(u)]^2 => u'=3(x^2)[cos(u)]^2
[sec(u)]^2 du=3x^2dx
兩端積分得 tan(u) =x^3+c => tan(y/x) =x^3+c
y(1)=0 => 0=1+c =>c=-1
tan(y/x)=x^3-1為隱函數解
--
Logic can be patient for it is eternal. ----- Oliver Heaviside
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