求特解 xy'+y=yln(xy)求通解上边那求通解y''-ay'^2=0y(0)=0,y'(0)=-1求特解怎么没人答呢。
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/21 09:21:42
![求特解 xy'+y=yln(xy)求通解上边那求通解y''-ay'^2=0y(0)=0,y'(0)=-1求特解怎么没人答呢。](/uploads/image/z/12075044-68-4.jpg?t=%E6%B1%82%E7%89%B9%E8%A7%A3+xy%27%2By%3Dyln%28xy%29%E6%B1%82%E9%80%9A%E8%A7%A3%E4%B8%8A%E8%BE%B9%E9%82%A3%E6%B1%82%E9%80%9A%E8%A7%A3y%27%27-ay%27%5E2%3D0y%280%29%3D0%EF%BC%8Cy%27%280%29%3D-1%E6%B1%82%E7%89%B9%E8%A7%A3%E6%80%8E%E4%B9%88%E6%B2%A1%E4%BA%BA%E7%AD%94%E5%91%A2%E3%80%82)
求特解 xy'+y=yln(xy)求通解上边那求通解y''-ay'^2=0y(0)=0,y'(0)=-1求特解怎么没人答呢。
求特解 xy'+y=yln(xy)求通解
上边那求通解
y''-ay'^2=0y(0)=0,y'(0)=-1求特解
怎么没人答呢。
求特解 xy'+y=yln(xy)求通解上边那求通解y''-ay'^2=0y(0)=0,y'(0)=-1求特解怎么没人答呢。
xy' + y = yln(xy)
令t = xy,dt/dx = y + x•dy/dx
dy/dx = (1/x)(dt/dx - t/x)
x(1/x)(dt/dx - t/x) + t/x = (t/x)ln(t)
dt/dx = (t/x)ln(t)
dt/(tlnt) = (1/x) dx
d(lnt)/lnt = (1/x) dx
ln|lnt| = ln|x| + lnC₁ = ln|C₁x|
lnt = C₁x
t = e^(C₁x) = C₁e^x
xy = C₁e^x
_______________________________________
y'' - a(y')² = 0,y(0) = 0,y'(0) = -1
令t = dy/dx,dt/dx = d²y/dx²
dt/dx - a•t² = 0
dt/dx = a•t²
dt/t² = a•dx
-1/t = ax + C₁
t = -1/(ax + C₁),y'(0) = -1 t(0) = -1
-1 = -1/C₁ => C₁ = 1
dy/dx = -1/(ax + 1)
y = (-1/a)ln(ax + 1) + C₂,y(0) = 0,a≠0
0 = (-1/a)ln(0 + 1) + C₂ => C₂ = 0
∴特解为y = (-1/a)ln(ax + 1)