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Exchange integration order ∫ (1,0) DX ∫ (x, 0) f (x, y) dy + ∫ (2,1) DX ∫ (2-x, 0) f (x, y) dy
∵ according to the mapping analysis of the upper and lower limits of the integral, the integral region is a triangle surrounded by straight lines y = x, x + y = 2 and y = 0
∴∫(1,0)dx∫(x,0)f(x,y)dy+∫(2,1)dx∫(2-x,0)f(x,y)dy=∫(1,0)dy∫(2-y,y)f(x,y)dx.
Higher number: general solution of differential equation dy / DX = Y / x + Tan (Y / x)
Let u = Y / x,
Then y = Xu
dy/dx=u+xdu/dx,
So the original equation becomes
u+xdu/dx=u+tanu,
xdu/dx=tanu,
du/tanu=dx/x
cosudu/sinu=dx/x
d(sinu)/sinu=dx/x
Integral on both sides
Ln|sinu| = ln|x| + C1, C1 is any real number,
sinu=(+,-)e^C1*x
Let C = (+, -) e ^ C1, then
sinu=Cx
u=arcsin(Cx)
y/x=u=arcsin(Cx)
y=xarcsin(Cx).
Urgent. Dy / dx-y / x = LNX 1. Dy / dx-y / x = LNX, y (1) = 2, SiNx + cosx ^ 3, 0 to π / 2 definite integral 3, cos (root x) 0 to π ^ 2 definite integral
1.y=e^(∫1/xdx)(∫lnx·e^(-∫1/xdx)dx+c)=x(∫lnx/xdx+c)=x(∫lnxdlnx+c)=x【(lnx) ²/ 2 + C] 2. Original formula = 1 + 2 / 3 = 5 / 33. Original formula = ∫ (0, π) ²) Cos √ xdx order √ x = TX = t ², DX = 2tdt, so the original formula = ∫ (0, π) cost · 2tdt = 2 ∫ (0, π) TDS
For the detailed explanation of the high number problem, y = y (x) is determined by √ (x ^ 2 + y ^ 2) = a * e ^ arctg (Y / x), and dy / DX is obtained My result is dy / DX = (2x (y ^ 2 + x ^ 2) + A ^ 2 * e ^ arctg (Y / x)) / (x * a ^ 2 * e ^ arctg (Y / x) - 2Y (y ^ 2 + x ^ 2)) If correct, do you need to simplify this step? Is there any requirement to simplify the final result of this problem?
Don't simplify!
Solving dy / DX in simple higher mathematics X ^ y = y ^ x, y is the function of X to find dy / DX. The first step is e ^ (ylnx) = e ^ (xlny). My first step is ylnx = xlny, which is the same later. Both sides find the derivative of X, but why is the answer different
The answer is different in form, but it can be Mutualized by the original equation x ^ y = y ^ x, so it is essentially the same
The derivative of the implicit function determined by the equation, and the characteristics of the result: 1. It generally contains dependent variables; 2. The result is not unique in form, that is, there are many forms of results
Find DX / dy x=3t^2 Y = sin4t, find DX / dy
dx/dt=6t
dy/dt=4cost
dy/dx=(dy/dt)*(dt/dx)=(4cost)*(1/6t)=2cost/3t
dx/dy=3t/2cost
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