I have a differential equation A*dx/dt + B(y-y0) = 0
Where x is a very complicated function of y.
How can I use Mathematica to rearrange y to get a function x in order to solve this?
Thanks
There are two or three different problems here that you might be asking:
Option 1: The subject line
First, if you really do have a function f[x] defined and you want to rearrange it, you would be doing something like this:
f[x_]=2+x+x^2;
Solve[y==f[x],x]
However, even here you should notice that inverse functions are not necessarily unique. There are two functions given, and the domain of each is only for y>=7/4.
Option 2: Solving a DE
Now, the equation you give is a differential equation. That is not the same as "rearranging a function y=f[x] into x=g[y]" because there are derivatives involved.
Mathematica has a built-in differential-equation solver:
DSolve[a y'[t] + b (y[t] - y0) == 0, y[t], t]
That will give you a function (in terms of constants $a,b,y_0$) that is the answer, and it will include the unspecified constant of integration.
Your system seems to refer to two functions, x(t) and y(t). You cannot solve one equation for two variables, so it is impossible to solve this (Mathematica or otherwise) without more information.
Option 3: Rearranging an expression
As a third alternative, if you are trying to rearrange this equation without solving the differential equation, you can do that:
Solve[a x'[t] + b(y[t]-y0)==0,x'[t]]
This will give you $x'(t)$ in terms of the other constants and the function $y(t)$, but in order to integrate this (i.e. to solve the differential equation) you will need to know more about y[t].
Related
I would like to solve problems combining boolean and integer logic in linear arithmetic with a SAT/SMT solver. At first glance, Z3 seems promising.
First of all, is it at all possible to solve the following problem? This answer makes it seem like it works.
int x,y,z
boolean a,b,c
( (3x + y - 2z >= 10) OR (A AND (NOT B OR C)) OR ((A == C) AND (x + y >= 5)) )
If so, how does Z3 solve this kind of problem in theory and is there any documentation about it?
I could think of two ways to solve this problem. One would be to convert the Boolean operations into a linear integer expression. Another solution I read about is to use the Nelson-Oppen Combination Method described in [Kro 08].
I found a corresponding documentation in chapter 3.2.2. Solving Arithmetical Fragments, Table 1 a listing of the implemented algorithms for a certain logic.
Yes, SMT solvers are quite good at solving problems of this sort. Your problem can be expressed using z3's Python interface like this:
from z3 import *
x, y, z = Ints('x y z')
A, B, C = Bools('A B C')
solve (Or(3*x + y - 2*z >= 10
, And(A, Or(Not(B), C))
, And(A == C, x + y >= 5)))
This prints:
[A = True, z = 3, y = 0, B = True, C = True, x = 5]
giving you a (not necessarily "the") model that satisfies your constraints.
SMT solvers can deal with integers, machine words (i.e., bit-vectors), reals, along with many other data types, and there are efficient procedures for combinations of linear-integer-arithmetic, booleans, uninterpreted-functions, bit-vectors amongst many others.
See http://smtlib.cs.uiowa.edu for many resources on SMT solving, including references to other work. Any given solver (i.e., z3, yices, cvc etc.) will be a collection of various algorithms, heuristics and tactics. It's hard to compare them directly as each shine in their own way for certain sublogics, but for the base set of linear-integer arithmetic, booleans, and bit-vectors, they should all perform fairly well. Looks like you already found some good references, so you can do further reading as necessary; though for most end users it's neither necessary nor that important to know how an SMT solver internally works.
I'm puzzled by what I think is a mistake in a partial derivative I'm having Mathematica do for me.
Specifically, this is what I have:
Derivative I'd like to take
I'm trying to take the partial derivative of the following w.r.t. the variable θ (apologies for the formatting):
f=(1/4)(-4e((1+θ)/2)ψ+eN((1+θ)/2)ψ+eN((1+θ)/2-θd)ψ)-s
But the solution Mathematica produces seems very different from the one I get when I take the derivative myself. While Mathematica says the partial derivative of f w.r.t. θ is:
(1/4)eψ(N-2)
By hand, I get and am quite confident the correct answer is instead:
(1/4)eψ(N(1-d)-2)
That is, Mathematica is producing something that drops the variable d when it is differentiating. I've explored different functions that take a derivative in Mathematica, and the possibility that maybe some of the variables I'm using (such as d) might be protected or otherwise special, but I can't say that I know why the answer's so off. This is the first time in the notebook that d appears, so it is not set to 0. For context, I'm trying to confirm that the derivative of the function is positive for values of the variables in certain ranges, and we have d>0 and d<(1/2). Doing this all by hand works but I'm trying to confirm with Mathematica as I will be dealing with more complicated functions and need to make sure I'm having Mathematica produce the right derivatives.
Your didn't add spaces in eN and θd, so it thinks they're some other 2-character variables.
Adding spaces between them gives your expected result:
f[θ,e,N,ψ,d,s] = (1/4) (-4 e ((1+θ)/2) ψ + e N ((1+θ)/2) ψ + e N ((1+θ)/2 - θ d) ψ) - s;
D[f[θ, e, N, ψ, d, s], θ] // FullSimplify
(* 1/4 e (-2 + N - d N) ψ *)
I HAVE EQUATION
y - 7(e^x/x)dydx=0
How to find analytic solution in Mathematica?
My work :
I simplify the equations become y'=yx/7e^x
I run in Mathematica,
DSolve[y'[x] == (y[x] x)/(7 e^x), y[x], x]
I get result:
{{y[x] -> E^(1/7 E^-x (-1 - x)) C[1]}}.
Questions:
Are my simplifications correct?
Do I type correct code in step 2?
How to find exact value of C in the result, because I want to use to find y' for given x value
Thank you for the answer
I'm an occasional Mathematica user and I am trying to transform an expression from spherical to Cartesian coordinates.
My function is defined as:
g[theta_, phi_] := Cos[phi](Sin[theta])^2 Sin[phi]
I'm hoping to transform that function using the following rules:
Sin[theta]Sin[phi] -> x
Cos[theta]-> y
Sin[theta]Cos[phi]-> z
in order to get the result:
zx
Here is the code I'm using to do that:
g[theta, phi] //. {Sin[theta]Sin[phi] -> x, Cos[theta] -> y, Sin[theta] Cos[phi] -> z}
And the result I get is:
Cos[phi] Sin[phi] Sin[theta]^2
So no transformation occurred.
Is there a function or an option I could add to help Mathematica figure out that the transformation is possible?
Thanks!
Perhaps this will be sufficient
Assuming[Sin[theta]Sin[phi]==x&&Cos[theta]==y&&Sin[theta]Cos[phi]==z,
Simplify[Cos[phi]Sin[theta]^2 Sin[phi]]]
which instantly returns
x z
That doesn't show you the steps or rules it used to arrive at that result, but because it considered x z to be "simpler" than your trig expression the evaluation process went in that direction.
There is a slightly more compact way of doing the same thing, if that matters. Simplify can accept a second argument which are the things which are assumed to be true during the process of simplification. Thus
Simplify[Cos[phi]Sin[theta]^2 Sin[phi],
Sin[theta]Sin[phi]==x&&Cos[theta]==y&&Sin[theta]Cos[phi]==z]
will give you exactly the same result
I have a source file like (without loss of generality (only to image a possible syntax)):
function a()
return g // global variable without any internal structure exactly
end
function b(x, y)
local z = x * y
return z + 1
end
function c(z, t)
return b(z * z, a())
end
// ...etc
I want to defferentiate any function WRT to some variable.
All the formal parametres we can treat as a functions with unknown at derive time internal structure.
If I stand correct further, then the following is truth (for depending symbols ' is part of symbol, for global variables is operator during substitute time stage (def: g{g} is one, but g{y} is zero)):
function a'()
return g';
end
function b'(x, y, x', y')
local z' = x' * y + x * y'
return z' + 0
end
But what to do with last function? Namely, with actual parameters in substitution of function b?
Is there any ready to use implementations of general algorithm to work with the above? What to do with higher order derivatives (especially interesting, how to handle the formal parameters)? Are there any other possible unclear cases?
I would suggest having your parameters be symbolic expressions that know how to respond to derivatives, and having all operations take functions and return functions. Then you will get a final expression that knows how to be represented as a derivative. Furthermore you can do things like partial derivatives at a later point because you have the symbolic expression.
For a real example of what I mean, see http://www.elem.com/~btilly/kelly-criterion/js/advanced-math.js for a library that I wrote to solve a calculus problem in JavaScript, and search for "Optimize if requested" in the source for http://www.elem.com/~btilly/kelly-criterion/betting-returns2.html to see how I used it. See http://www.elem.com/~btilly/kelly-criterion/ for an explanation of why I was writing that code.
In that example I, of course, was not working from infix notation. But that is a standard parsing problem that I think you know how to solve.