dynamic – how to solve the two differential equations?

i want to solve the following equation to find k at r max is 15,17 and 19 and plot tension(kx) vs angle(theta) there have any way to solve if it have many unknows?

mgSin((Theta)(t))) – (m(r”(t) – r(t) (((Theta)'(t))^2))) ==
k (r(t) – 14) —-1

gCos((Theta)(t)) == r(t)(Theta)”(t) + 2r'(t)(Theta)'(t) ——2

m=81 g=9.81 r0=15 theta0=arcsin((r max(15,17,19)-12)/15)

thank you very much.

How to solve non-linear equations coming out of lagrange multiplier?

Using Lagrange multipliers I obtained the following system of equations:

begin{align*}x+y+z &= 20\
x^2 + y^2 + z^2 &= 200\
yz &= lambda + 2 mu x\
xz &= lambda + 2 mu y\
xy &= lambda + 2 mu z
end{align*}

I am struggling to solve this system of equations. I have managed to relate $lambda$ and $mu$. First I square the first equation and substitute the second equation:

$$(x+y+z)^2 = 200 + 2xy + 2xz + 2yz = 400 $$

$$xy+xz+yz = 100 $$

Then I add the last three equations

$$xy+xz+yz=3lambda+2mu(x+y+z) =3lambda + 40 mu=100$$

$$3 lambda=100 -40mu $$

But then I am stuck.
I have confirmed that there are 6 distinct solutions to this system. What is the strategy when trying to solve this kind of system of equations?

database design – How to break down MySQL big table into multiple to solve operational issue

This is my current table schema in MySQL
we are not facing any performance issue in our application but there are operational issue .
The size of this table is 7 TB (6TB data and 1 TB index ) and has 4 Billions rows .

Because of the this one table having this big size we are not able to do any alter table on this table .
We have to use percona which takes 1 week to complete .

so to handle this we have decided to break this table in to 3 .
There are two columns which stores xml file that we want to move to separate table and this two column alone takes 2.5 TB of storage .

DETAILS longtext, and SUMMARY varchar(4000) DEFAULT NULL,

Along with it we also want to move few more columns to another table so that all three tables will become lighter
like third table we want to move USES_TYPE varchar(255) NOT NULL, STEP_TYPE varchar(255) NOT NULL, NAME varchar(1500) DEFAULT NULL, and REMARKS varchar(1000) DEFAULT NULL,

CREATE TABLE `app_data` (
`ID` varchar(255) NOT NULL,
`USES_TYPE` varchar(255) NOT NULL,
`STEP_TYPE` varchar(255) NOT NULL,
`CUST_ID` varchar(255) DEFAULT NULL,
`DETAILS` longtext,
`DATE_TIME` datetime(6) DEFAULT NULL,
`GROUP_ID` varchar(255) DEFAULT NULL,
`SYSTEM_ID` varchar(255) DEFAULT NULL,
`NAME` varchar(1500) DEFAULT NULL,
`CUSTOMER_ID` varchar(255) DEFAULT NULL,
`REMARKS` varchar(1000) DEFAULT NULL,
`SUMMARY` varchar(4000) DEFAULT NULL,

PRIMARY KEY (`ID`),
KEY `IDX_APP_DATA_CID_OT` (`CUST_ID`,`USES_TYPE`) USING BTREE,
KEY `IDX_APP_DATA_SYSTEM_ID` (`SYSTEM_ID`) USING BTREE,
) ENGINE=InnoDB DEFAULT CHARSET=utf8mb4 COLLATE=utf8mb4_bin;

Now my final table will be comething like 

Table one 

CREATE TABLE `app_data_table1` (
`ID` varchar(255) NOT NULL,
`DETAILS` longtext,
`SUMMARY` varchar(4000) DEFAULT NULL;
)

Table Two would be like 

CREATE TABLE `app_data_table2` (
`ID` varchar(255) NOT NULL,
`USES_TYPE` varchar(255) NOT NULL,
`STEP_TYPE` varchar(255) NOT NULL,
`NAME` varchar(1500) DEFAULT NULL,
`REMARKS` varchar(1000) DEFAULT NULL;
)

and table three 

CREATE TABLE `app_data` (
`ID` varchar(255) NOT NULL,
`CUST_ID` varchar(255) DEFAULT NULL,
`DATE_TIME` datetime(6) DEFAULT NULL,
`GROUP_ID` varchar(255) DEFAULT NULL,
`SYSTEM_ID` varchar(255) DEFAULT NULL,
`CUSTOMER_ID` varchar(255) DEFAULT NULL,

PRIMARY KEY (`ID`),
KEY `IDX_APP_DATA_CID_OT` (`CUST_ID`) USING BTREE,
KEY `IDX_APP_DATA_SYSTEM_ID` (`SYSTEM_ID`) USING BTREE,
) ENGINE=InnoDB DEFAULT CHARSET=utf8mb4 COLLATE=utf8mb4_bin;

I am new to data base but this is what i am coming up i know this is not optmised so my humble request is to pleae guide me on this.

Once we do this we need to use join to dispaly on UI or where ever we fetch so will that be slower ?

trigonometry – Solve limit for a

Determine a so that: $lim_{xtoinfty} frac{tan(ax)}{sin(x)} = 2$

So far, I have used the L’hopital rule:

$frac{frac{1}{a cos(x)}}{cos(x)} = frac{1}{a cos^3(x)} = 2$

But I am not sure if this is the right way of solving this limit. Can anyone help me?

Solve Matrix equations with Cross Product: weird system of equations

I would like to find the values {Pfx, Pfy, Pfz} that satisfy the equation A X B = C . The code of everything is at the end, I want to ilustrate with images what i think of first:

A is this:

Matrix A

B is this: {0,0,0}

And C is this: {100,500,200}

The image of the complete code is this one:

Complete code

My variable here are {Pfx,Pfy,Pfz} and the {i,j,k} are the unit vectors. The way that the matrix is shown in the picture corresponds with a trick used to solve in papper this type of matrix.

The solution would give me the value of Pfx in the “x” coordinate (and this would be expressed by Pfx being multiplied with the i vector). And the same mechanism applies with Pfy related with j Vector ; and Pfz related with k.

The problem here comes with the fact that the I can`t find the values {Pfx, Pfy, Pfz} that satisfy the equation A X B = C.
I am not sure if the problem lies in the “LinearSolve” comand, in the use of the CrossProduct or in the use of the versor {i, j, k} inside the matrix.

Any kind of help in this regard will be extreamly useful, thanks in advance!!

Code:

i := {1, 0, 0}

j := {0, 1, 0}

k := {0, 0, 1}

LinearSolve(({ {i, j, k},{1, 2, 3},{Pfx, Pfy, Pfz}})(Cross)({{0},{0},{0}}) == ( {{100},{500},{200}} )) 


equation solving – Why is Reduce unable to solve this system of inequalities?

I need to solve the following inequalities but the execution just keeps going..

Reduce((1 - 
       beta)*(-(1/lamda)*
        Log((delta - beta*(1 - 2 d))/(lamda*delta)))^(rho - 
        1) - (alpha*
       A^rho/d^2) (1 - (1/(lamda*beta*d*(1 - d)))*
         Log((delta - beta (1 - 2 d))/(lamda*delta)))^{alpha*rho - 
        1} - lamda*(delta - 
       beta (1 - 2 d)) (beta*
         A^rho (1 - (1/(lamda*beta*d*(1 - d)))*
             Log((delta - beta (1 - 2 d))/(lamda*delta)))^{alpha*
            rho} + (1 - 
           beta)*(-(1/lamda)*
            Log((delta - beta (1 - 2 d))/(lamda*delta)))^
          rho)/(beta*(1 - 2 d)) < 0 && 
  delta - beta*(1 - 2 d) > 
   0 && (delta - beta*(1 - 2 d))/(lamda*delta) < 1 && rho < 1 && 
  A > 0 && 0 < alpha < 1 && 1 > beta > 0 && 1 > lamda > 0 && 
  delta > 0 && 0 < d < 1/2, beta, Reals)

Is it a problem with Reduce ?

differential equations – How to solve the coupled PDEs by NDSolve?

I want to solve the following PDEs numerically by NDSolve.
enter image description here

pde = {D((Alpha)((Tau), X, Y), (Tau)) - 
I*(Kappa)(X, Y)*(Alpha)((Tau), X, Y) == 
   (D((Alpha)((Tau), X, Y), X, X) + 
  D((Alpha)((Tau), X, Y), Y, Y)) + (1 - (Beta)((Tau), X, Y) - 
   Abs((Alpha)((Tau), X, Y))^2)*
       (Alpha)((Tau), X, Y), 


Subscript((Tau), o)*D((Beta)((Tau), X, Y), (Tau)) == 
       (Subscript(D, d)/(Xi)^2)*(D((Beta)((Tau), X, Y), X, X) + 
       D((Beta)((Tau), X, Y), Y, Y)) + 
         (Subscript((CurlyPhi), o)^2/(Subscript(T, c)*
         Subscript((Rho), n)*Subscript(C, v)))*
       
 Grad((Kappa)(X, Y), {X, Y}) . Grad((Kappa)(X, Y), {X, Y}) - 
     ((Beta)((Tau), X, Y) - Subscript(T, b)/Subscript(T, c))/
 Subscript((Tau), eph), 
 Laplacian((Kappa)(X, Y), {X, 
 Y}) == (Subscript((Psi), o)^2/(4*e*Subscript(k, B)*
    Subscript(T, c)*
           Subscript((CurlyPhi), o)^2))*
Div(Im(ConjugateTranspose((Alpha)((Tau), X, Y))*
   Grad((Alpha)((Tau), X, Y), {X, Y})), 
       {X, Y})}
bc = {(Beta)(0, X, Y) == Subscript(T, b)/Subscript(T, c), 
     (Beta)(0, 0, 0) == (Subscript(T, b) + dT)/Subscript(T, c), 
     (Beta)((Tau), l/2, Y) == 
   Subscript(T, b)/Subscript(T, c), (Beta)((Tau), -(l/2), Y) == 
       Subscript(T, b)/Subscript(T, c), 
  D((Beta)((Tau), X, Y), Y) == 0 /. Y -> w/2, 
     D((Beta)((Tau), X, Y), Y) == 0 /. 
   Y -> -(w/2), (Alpha)(0, X, Y) == 0.68, (Alpha)((Tau), l/2, Y) == 
   0, 
     (Alpha)((Tau), -(l/2), Y) == 0, 
  D((Alpha)((Tau), X, Y), Y) == 0 /. Y -> w/2, 
     D((Alpha)((Tau), X, Y), Y) == 0 /. Y -> -(w/2), 
  D((Kappa)(X, Y), X) == ((-Subscript((Rho), n))*Is)/(w*d) /. 
       X -> l/2, 
  D((Kappa)(X, Y), X) == ((-Subscript((Rho), n))*Is)/(w*d) /. 
   X -> -(l/2), 
     D((Kappa)(X, Y), Y) == 0 /. Y -> w/2, 
  D((Kappa)(X, Y), Y) == 0 /. Y -> -(w/2)}

Here 𝛼 and 𝛽 are functions of 𝜏, X, Y, and 𝜅 is the function of X, Y.
However, the result shows there are inputs in error. (I use Mathematica 9.0)
Is that any problem in the setting of boundary conditions and initial values?

enter image description here

What is the problem?
The related parameters are:

Subscript(n, o) = 100; 
Subscript((Rho), n) = 1; 
Is = 10; 
l = 80; 
w = 20; 
d = 5; 
dT = 3; 
Subscript(T, b) = 7.5; 
Subscript(T, c) = 15; 
Subscript((Tau), eph) = 0.0001; 
Subscript((Tau), o) = 0.01; 
e = 1; 
Subscript(k, B) = 1; 
Subscript((CurlyPhi), o) = 1; 
Subscript((Psi), o) = 1; 
Subscript(C, v) = 5; 
(Xi) = 4.5; 
Subscript(D, d) = 0.5;

or

enter image description here

How to solve core web vital in search console?

How to solve core web vital in search console? – Webmasters Stack Exchange

Solve a cubic equation of one variable with parameters!

x^3 + Cos((Theta))x^2 + E^(I(Theta))Cos((Theta))x + E^(
I
(Theta))=0
and
x^3 + I
Sin((Theta))x^2 + E^(I(Theta))ISin((Theta))x – E^(
I
(Theta))=0
They come from these two matrices, and I’m trying to figure out its eigenvectors, but I can’t figure out its eigenvalues.
( {
{0, 0, E^(I*(Theta))},
{-Cos((Theta)), -ISin((Theta)), 0},
{-I
Sin((Theta)), -Cos((Theta)), 0}
} )
and
( {
{-Cos((Theta)), -ISin((Theta)), 0},
{0, 0, E^(I
(Theta))},
{-I*Sin((Theta)), -Cos((Theta)), 0}
} )

Both equations are written in the corresponding code, which can be opened with the appropriate mathematical software to view the original format.
Thanks!

Solve for pi; showing a proof for logistic regression

I need help solving for π and am extremely confused! I know I have to use the base e function but confused how to get there!

ln(π/1-π)=B0+B1x1
Solve the equation for π to show π=exp⁡{B0+B1x1}/(1+exp⁡{B0+B1x1})

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