Differential equation 2ydx + (y ^ 2-6x) dy = 0

Differential equation 2ydx + (y ^ 2-6x) dy = 0

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The general solution of differential equation 2ydx + (y ^ 2-6x) dy = 0 (hint: take x as the function of Y) is solved
Find the general solution of the differential equation........

We can't use the method of prompt. Other methods can be reduced to 2 (YDX XDY) / y ^ 2 = (3x / y ^ 2-1) dy2d (x / y) = (4x / y ^ 2-1) dy. let u = x / y, reduce to 2du = (4u-1) dyd (LN (4u-1)) = 2dy4u-1 = cexp (2Y) C be any constant. If u = x / y is substituted, 4x / Y-1 = cexp (2Y), I won't simplify any more

Verify that the function y = CE ^ (- x) + X + 1 is the general solution of the differential equation y '= y + X, and find the special solution satisfying the initial condition y | (x = 0) = 2,

The general solution of the differential equation y '= y + X is y = CE ^ (x) - X-1
Because: y = CE ^ (x) - X-1, so y '= CE ^ (- x) - 1, so: y' = y + X,
So the general solution of the differential equation y '= y + X is y = CE ^ (x) - X-1
Because y | (x = 0) = 2, substituting C = 3, satisfying the initial condition y | (x = 0) = 2, the special solution is y = 3E ^ (x) - X-1

Let ∑ UN be absolutely convergent and ∑ VN be convergent, then ∑ UN VN be absolutely convergent

To prove the absolute convergence of ∑ unvn | is to prove that the series ∑ unvn | = ∑ UN | VN | converges. Because ∑ VN converges, the sequence {VN} is bounded (because limvn = 0), so | VN | ≤ M

The boundedness proof of convergent sequence
Let Lim xn = A and E = 1, then there is n > 0. When n > N, there is / xn-a/

That's from the definition of LIM xn = a
By using the definition of limit, we can get | xn if we start N and then all (here are infinite) xn are bounded|

What does it mean to construct a sequence UN related to SN
sn=(1+n)n/2
What is un

SN is a sum sequence
UN is a general term
un=Sn+1-Sn
Sn=(1+n)n/2
un=n

If the partial sum of an infinite series Sn = [(2 ^ n) - 1] / 2 ^ n is known, then the general term UN of the series
I don't understand the concept

Just know the meaning of partial sum
Economic Mathematics team to answer for you, there is not clear please ask. Please timely evaluation

It is known that the series ∑ (UN) ^ 2 ∑ (VN) ^ 2 converges, and it is proved that ∑ (UN + VN) ^ 2 converges
If you use absolute convergence, say the usage of absolute convergence here

（un+vn）^2=(un)^2 +2unvn+(vn)^2《(un)^2 +2|unvn|+(vn)^2《2[(un)^2 +(vn)^2]
The series ∑ (UN) ^ 2 ∑ (VN) ^ 2 converges, so the series 2 [(UN) ^ 2 + (VN) ^ 2] converges, so the series ∑ (UN + VN) ^ 2 also converges according to the comparison criterion

If the limit UN is equal to a, what does the series (un-un-1) converge to?

The series (un-un-1) converges to 0

It is proved that if the positive series ∑ UN converges, then ∑ UN / (1 + UN) also converges

If the series UN converges, then UN converges to 0. Therefore, when n tends to infinity, UN / (1 + UN) is equivalent to UN, and both of them converge and disperse. Therefore, the new series converges