Questions tagged [algebraic-manipulation]

The art of manipulating an algebraic expression into the desired form.

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24 views

Easy way to extract coefficients of functions in an ODE/PDE?

Suppose I have an ODE like $$ \epsilon y''(x) + xy'(x) + \epsilon^2 y(x)=0$$ or a PDE like $$ \dfrac{{\cal U}}{{\cal T}}\dfrac{\partial u}{\partial t}+\dfrac{{\cal U}^{2}}{{\cal L}}u\dfrac{\partial u}{...
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3answers
151 views

Verify that given expr is a general solution to a given ODE

As in the title, for example: eq = y''[x] + 4 y[x] == 7; sol = DSolveValue[{eq}, y[x], x] 7/4 + C[1] Cos[2 x] + C[2] Sin[2 x] ...
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71 views

How to sum the KroneckerDelta[] in equation?

How to sum the KroneckerDeltas in following equation? k, p1, ...
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2answers
53 views

How to efficiently simplify using PowerExpand

I am tying to simplify the following expression using Simplify, FullSimplify, Expand and <...
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0answers
28 views

Simplifying Expressions in a Different Way

Let's say I have the following code, f[x_]:=(b-x)/(b-a) (b-a)*f[x]^2 Is there a way to get Mathematica to output (b-x)*f[x] ...
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1answer
61 views

Taking absolute square of the following term

I have this equation and I wanted to find the absolute square of this term in Mathematica $$ H_1 = \left | 1 + j\; \left(\frac{ 2 \pi}{1-e^\lambda}\right) \right|^2 $$ I did this in mathematica <...
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1answer
70 views

Is it possible to ask Mathematica to separate all the terms containing $x$ and $y$ in the given function?

I have a 6-variable function like this $ f(x,y,z,t,w,v)= 8 t^6 e^{i (3 t v+2 t w-x+2 y)}+t^6 e^{i (2 t v-x+y)}-2 t^5 e^{i t (v+3 w)}+2 t^5 e^{4 i t w}+3 t^4 e^{i t (v+w)}+2 t^4 e^{2 i t w}+2 e^{i (t (...
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2answers
94 views

Computing the sign of an expression

Can you please help with the following. I would like to compute the sign of an expression: ...
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1answer
28 views

Perform Eliminate, Solve and Format at once in Mathematica

I have a series of equations where I use eliminate to get a new equation with desired variables. This equation is then solved for a specific variable and then the result is presented in a specific ...
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1answer
61 views

Extracting coefficients from equation [closed]

I am looking for a Mathematica code that finds the coefficients from an equation. So for example, when I have the equation $ax^2+bx+c$ I want to the code to spit out: ...
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4answers
85 views

numerical values of coefficients

Consider the following output: x= (Sqrt[3/(2 (1 + Sqrt[11]/4))] b[0, π/3] - (1 + Sqrt[7/3]) b[(2 π)/3, (2 π)/3])/(4 Sqrt[2] 7^(1/4) Sqrt[1 + Sqrt[7]/4]) If i do <...
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1answer
61 views

Mathematica Partial Fraction Decomposition full steps [closed]

I hope to get full steps instead of getting the result. Any ways to do so?
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2answers
118 views

How to factor a polynomial expression with variable coefficients (and imaginary values)

How can I factor terms with variable coefficients? For example for the expression: $x^2 + a$ I know that I can factor this to: $(x+i\sqrt{a})(x -i\sqrt{a})$ How do I do this in mathematica? I would'...
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2answers
57 views

Collect terms in inverse powers of three variables

I have the following expression ...
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2answers
57 views

Solve for two variables but output second variable in terms of first variable

Here are some equations for illustration: Format[Solve[{4*r1^3 == r2^2*(3*h), t*r2^2*(h - d) == r2^2*(h + d)}, #] & /@ {{h, d}}, InputForm] Here is their ...
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1answer
81 views

RootApproximant for a function

DSolve yield this rather lengthy solution ...
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1answer
70 views

How to convert trigonometric values to AlgebraicNumber

It is known that trigonometric functions evaluated at rational numbers of degrees are algebraic, and Mathematica recognize this: for example ...
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1answer
48 views

TensorExpand, TensorProduct and Distribution over Scalars

Consider: TensorExpand[( b1 x )\[TensorProduct] (b1 x + b2), Assumptions -> x ∈ Reals] (*x^2 b1\[TensorProduct]b1 + x b1\[TensorProduct]b2*) This works as ...
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1answer
86 views

Resolve Error or Mathematica's Bug?

The following code, Resolve[Exists[{n1, n2, n3}, {n1, n2, n3} \[Element] NonNegativeIntegers, 1/2 (1 + n1 + n2 - n3) \[Element] NonPositiveIntegers]] ...
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1answer
77 views

Simple Question: How to simplify what is inside the Sin function like below?

I have this code of Mathematica (1/(2 (fj - fk) π)) Cos[(fj - fk) π T + (fj - fk) π (T + δt)] Sin[(fj - fk) π T - (fj - fk) π (T + δt)] There are a few ...
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2answers
49 views

Solve the system using elimination What are a, b, and c in the quadratic equation, ax^2+bx+c=y given the [closed]

Solve the system using elimination What are a, b, and c in the quadratic equation, ax^2+bx+c=y given the points: (2,17), (-2,9), and (1,6), please solve the question using steps e.g. (step 1, step 2)?
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1answer
35 views

Changing initial conditions in a recursive polynomial family definition. Then, finding the generation function for this family

I'm following a paper trying to find a way to repeat the done computations using Mathematica. I'm beginner using Mathematica and I already read that: I could define functions by recursion; I could ...
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39 views

Testing Matrix inequalities

I was wondering how one might check whether matrix inequalities are true/false the same way it can be done for scalar variables (like in this post). Say I want to test the obvious inequality A,B are ...
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1answer
66 views

Counting terms in recursive operation

Suppose $X$ is an algebra and $T :X\to X$ is linear function. Let $L:X\to X$ be a function satisfying the following property $$L(a \cdot b)= T(a)\cdot b + a \cdot T(b)+b\cdot T(a)+T(b)\cdot a\tag{*}$$...
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2answers
114 views

Commutation, symmetrizer and duplication matrices

The following 3 matrices are useful when viewing matrices as vectors, known as commutation $K_n$, symmetrizer $N_n$ and duplication $G_n$. They are usually defined by their matrix relations below. $$ \...
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2answers
251 views

How to write a positive function as sum of positive terms only

A multivariate polynomial function can be written in a certain form (x^m (1-x)^n y^k + ... ) if the function is Positive in the whole domain and contains a finite ...
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4answers
315 views

How to generate all combination of multiplication of multiple variables (raised to some powers)

I want to generate all the possible combinations (commutative) of a few variables but also raised to some fixed powers. Lets take the following example: I have three variables ...
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2answers
91 views

How to simplify an equation corresponding to an ellipsoid? [duplicate]

The equation 29/36 - x/2 + x^2/4 + y/9 + y^2/36 - (4 z)/9 + z^2/9 == 1 is equivalent to ...
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2answers
82 views

Inverse function and Integral: How to simplify this DSolve output?

I'm trying to DSolve these equations with each other: ...
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1answer
53 views

How to compute the factors in Bezout's Identity

If I have two polynomials $p(x),q(x)$, PolynomialGCD[p,q,x] can give me their greatest common divisor. But how to compute the polynomials $a(x),b(x)$ such that $d(x)...
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1answer
51 views

How do you convert an ordinary equation into a parametric equation? [duplicate]

It's a 3D linear equation {P1: 2x-y-3z+2=0,P1: x+2y-z-6=0} Are there any methods that can convert this linear equation to its parametric form. Ps:the parametric form is {x=7t, y=-t+14/5, z=5t}
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1answer
52 views

How to use Collect to group negative terms?

For example, I have: $-d^2-k^{d+1}+d k^d+d+k-1$. I want to get: $-\left(d^2+k^{d+1}+1\right)+d k^d+d+k$.
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1answer
72 views

Solving a coupled system of ODEs

I have two coupled ODEs for $T(x)$ and $t(x)$: $$\frac{d^2 T}{d x^2}-\beta (T-t)+K=0 \tag 1$$ $$\frac{d t}{dx}-\alpha(T-t)=0 \tag 2$$ $\alpha, \beta$ and $K$ are constants $>0$. Also, it is known ...
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1answer
59 views

How to get the result in terms of a previously defined expression?

Say I have a vector X[n_] := Table[Subscript[x, i], {i, 1, n}] and a matrix of connections: ...
4
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2answers
78 views

Trouble verifying a solution to a differential equation

I am reading a physics book which has discussed one approach towards solving the differential equation $$ \frac{d^2 x(t)}{dt^2} = Cx(t) $$ as follows: Using Mathematica to solve the equation. I tried ...
2
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1answer
101 views

Numerical integration of MeijerG function with variables

I applied the integration, How to solve the function which is mentioned below. here, 'theta' is the only variable. q13, q1,q2,p,qi1 are all variables will take value from the loop ranges from 0 to 2, ...
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0answers
31 views

Symbolic Set Difference

Suppose I have two symbolic set containing 2-points like A={(m,n)| for m,n belonging to whole numbers}. So m,n=0,1,2,... and I have another set containing 2-points ...
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1answer
153 views

Derivative of real antisymmetric matrix in mathematica

Is it possible to find the derivative of components of a real antisymmetric matrix using index notation? Eg: I have a very large real antisymmetric matrix. Then from Matrix Cookbook, we know the ...
2
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1answer
85 views

Converting an equation to the form $ax+by=c$

I have a linear equation that comes as a result of applying orthogonality on a non-homogeneous boundary condition with two unknowns $C_1$ and $C_2$. Can someone help me to express this equation in the ...
4
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1answer
141 views

Simplifying the final coefficient expression of the solution to a Laplacian

I have been trying to solve a boundary value problem analytically which involves a three-dimensional temperature Laplacian over a parallelepiped. In the final step of my solution, using the two non-...
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1answer
121 views

Solving a system of polynomial equations - can I be sure that the number of solutions, and the solutions are correct?

I need to solve a system of two polynomials with integer coefficients in two variables, $\{Q_1(w,z)=0,\,Q_2(w,z)=0\}$, and want to compute all real solutions. I'm able to run ...
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1answer
67 views

Counting real roots of bivariate polynomial

For a proof, I'm working with a polynomial $p(x,m)$ and I need to show that: For any $m\in (0,6)$, $p_m(x)\equiv p(x,m)$ has a unique real root in $(0,1/2)$. I find that this is true by using Reduce ...
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0answers
57 views

How implement Dual Quaternion algebra in Mathematica?

How can we manipulate and operate with dual quaternionic variables in Mathematica? Since a dual quaternion can be expressed like this $\hat q = a + \epsilon b $ where $a,b \in \mathbb{H}$, or ...
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1answer
116 views

How to express this radical in terms of Golden Ratio?

(1/3)*Sqrt[(1/11)*(-3 - 2*Sqrt[5] + 3*Sqrt[178 + 82*Sqrt[5]])] $$ r=\frac{1}{3} \sqrt{-2 \sqrt{5}+3 \sqrt{82 \sqrt{5}+178} \over 11} $$ I need to express this ...
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1answer
48 views

Define trace-like function (e.g. cyclically invariant) involving operators without built-in meaning

I am interested in defining a function, say $\mathrm{Tra}$, that imply operators without build-in meaning, say it involves products with the $\cdot$ (Esc . Esc), ...
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1answer
42 views

Derive arbitrary steps between two expressions

Can someone show me the syntax to find the algebraic transformations between: $\sqrt{1+(\frac{dy}{dx})^2}dx$ and $\sqrt{dx^2+dy^2}$ And more generally, how to get the algebraic derivation between ...
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1answer
68 views

Resolving a sum after making a substitution

Consider the following sum as expressed by $s3$: ...
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3answers
74 views

Mathematica inverting algebraic output order

This is a somewhat silly requirement, but would help a lot in my workflow. Whenever I output an algebraic expression which contains a number, mathematica always orders the number first whenever ...
3
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1answer
51 views

Replacing an indexed term

Consider the expression: s[i_, n_] := Sum[(e[c[j]] + e[z[j]])*(l[i, j] + m[i, j]), {j, 1, n}] I'm trying to replace e[c[3]] by c[3] and have attempted ...
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0answers
34 views

Expanding complex expression and replacing variables in the result

When I'm finished expanding my expression I would like the result to be expressed again in terms of the original variables (like α, S1, T1, T2). I've tried replace, ...

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