Question

求微分方程yy''=3y'^2的通解，简述过程不要只给答案。谢谢

Answer 1

y * y'' = 3(y')²
(令u = y' = dy/dx，y'' = d²y/dx² = d(y')/dx = du/dx = du/dy * dy/dx = du/dy * u
y * du/dy * u = 3u²
y * du/dy = 3u
du/u = 3*du/y
ln(u) = 3ln(y) + C
u = e^[ln(y³) + C]
= y³ * C1
dy/dx = y³ * C1
dy/y³ = C1*dx
-1/(2y²) = C1*x + C2
2y² = -1/(C1*x + C2)
y² = -1/[2(C1*x + C2)]
= C3/(2x + C4)
弄漂亮点就是y² = C₁/(2x + C₂)
这里所有的常数项都是不同的

Answer 2

y’’/y'=3y'/y两边积分
lny'=3lny+c
下面不会了

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无语了

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