Equation Solving

The Wolfram Language encompasses the world's most extensive array of numerical and symbolic equation-solving tools, featuring numerous pioneering algorithms. These are all readily accessible via a limited set of highly efficient functions. Thanks to the symbolic framework of the Wolfram Language, both equations and their solutions can be easily represented symbolically, and seamlessly incorporated into computations and visualizations.

Solve

Address a quadratic equation.

Download original notebook

sol = Solve[(*SpB[*)Power[x(*|*),(*|*)2](*]SpB*) + a x + 1 == 0, x]

{{x->((*FB[*)((1)(*,*)/(*,*)(2))(*]FB*)) (-a-(*SqB[*)Sqrt[-4+(*SpB[*)Power[a(*|*),(*|*)2](*]SpB*)](*]SqB*))},{x->((*FB[*)((1)(*,*)/(*,*)(2))(*]FB*)) (-a+(*SqB[*)Sqrt[-4+(*SpB[*)Power[a(*|*),(*|*)2](*]SpB*)](*]SqB*))}}

Solve simultaneous equations in $x$ and $y$

Solve[a x + y == 7 && b x - y == 1, {x, y}]

{{x->(*FB[*)((8)(*,*)/(*,*)(a+b))(*]FB*),y->-(*FB[*)((a-7 b)(*,*)/(*,*)(a+b))(*]FB*)}}

Solve equations in a geometric region

Solve[
  Element[{x, y}, InfiniteLine[{{0, 0}, {2, 1}}]] && 
  Element[{x, y}, Circle[]]
, {x, y}]

{{x->-(*FB[*)((2)(*,*)/(*,*)((*SqB[*)Sqrt[5](*]SqB*)))(*]FB*),y->-(*FB[*)((1)(*,*)/(*,*)((*SqB[*)Sqrt[5](*]SqB*)))(*]FB*)},{x->(*FB[*)((2)(*,*)/(*,*)((*SqB[*)Sqrt[5](*]SqB*)))(*]FB*),y->(*FB[*)((1)(*,*)/(*,*)((*SqB[*)Sqrt[5](*]SqB*)))(*]FB*)}}

Graphics[{{Blue, Line[Normalize /@ {-{2, 1}, {2, 1}}], 
   Circle[{0,0}, 1]}, {PointSize[Large], Red, Point[{x, y}] /. %}}, AspectRatio->1, ImageSize->Small]

(*VB[*)(FrontEndRef["77e3851b-8470-4241-a0b0-9e2b32a33412"])(*,*)(*"1:eJxTTMoPSmNkYGAoZgESHvk5KRCeEJBwK8rPK3HNS3GtSE0uLUlMykkNVgEKm5unGluYGibpWpiYG+iaGJkY6iYaJBnoWqYaJRkbJRobmxgaAQBx8BSf"*)(*]VB*)

NSolve

Approximate solutions to a polynomial equation:

NSolve[(*SpB[*)Power[x(*|*),(*|*)5](*]SpB*) - 2 x + 3 == 0, x]

{{x->-1.4236058485523317`},{x->-0.24672925691056408`-1.3208163474502472` I},{x->-0.24672925691056408`+1.3208163474502472` I},{x->0.9585321811867298` -0.49842777903184593` I},{x->0.9585321811867298` +0.49842777903184593` I}}

Solve the equation $sin(z + sin(z + sin(z))) = cos(z + cos(z + cos(z)))$

eq[z_] := Sin[z + Sin[z + Sin[z]]] == Cos[z + Cos[z + Cos[z]]]
roots = NSolve[eq[z] && -3 < 
     Re[z] < 3 && -3 < Im[z] < 3, z, MaxRoots -> Infinity];

ListPlot[{Re[z], Im[z]} /. roots, 
 PlotLabel -> 
  ToString[Sin[z + Sin[z + Sin[z]]] == Cos[z + Cos[z + Cos[z]]]]]

(*VB[*)(FrontEndRef["f7a08193-ad1d-4419-b796-e757c01d1fb1"])(*,*)(*"1:eJxTTMoPSmNkYGAoZgESHvk5KRCeEJBwK8rPK3HNS3GtSE0uLUlMykkNVgEKp5knGlgYWhrrJqYYpuiamBha6iaZW5rpppqbmicbGKYYpiUZAgCAGhV0"*)(*]VB*)

FindRoot

Searches for a numerical root

Find the solution near 0

FindRoot[Cos[x] == x, {x, 0}]

{x->0.7390851332151607`}

Solve a nonlinear system of equations:

FindRoot[{Exp[x - 2] == y, y^2 == x}, {{x, 1}, {y, 1}}]

{x->0.019026016103714054`,y->0.13793482556524314`}

Compute inverse function

inv[f_, s_] := Function[{t}, s /. FindRoot[f - t, {s, 1}]]

Approximate exponent

einv = inv[Exp[x], x]
Plot[{einv[x], Log[x] + 0.5}, {x, 0, 1}, PlotRange -> All]

Function[{t$},x/. FindRoot[(*SpB[*)Power[E(*|*),(*|*)x](*]SpB*)-t$,{x,1}]]

(*VB[*)(FrontEndRef["d672b5b7-6311-4f00-82a3-d67392ad06f5"])(*,*)(*"1:eJxTTMoPSmNkYGAoZgESHvk5KRCeEJBwK8rPK3HNS3GtSE0uLUlMykkNVgEKp5iZGyWZJpnrmhkbGuqapBkY6FoYJRrrAsWNLY0SUwzM0kwBeK4VDg=="*)(*]VB*)

Find minimum

Find a local minimum, starting the search at $x=2$:

FindMinimum[x Cos[x], {x, 2}]

{-3.2883713955908966`,{x->3.425618459492147`}}

Plot[x Cos[x], {x, 0, 20}, Prolog->{Red, Line[{{0,0}, {20,0}}]}]

(*VB[*)(FrontEndRef["241d39f0-7923-42c1-b6dd-586077aebc69"])(*,*)(*"1:eJxTTMoPSmNkYGAoZgESHvk5KRCeEJBwK8rPK3HNS3GtSE0uLUlMykkNVgEKG5kYphhbphnomlsaGeuaGCUb6iaZpaTomlqYGZibJ6YmJZtZAgB2xxVK"*)(*]VB*)

Find a minimum of a function over a geometric region:

FindMinimum[{x + y, Element[{x, y}, Disk[]]}, {x, y}]

{-1.4142152367632073`,{x->-0.7071076183816036`,y->-0.7071076183816036`}}

Show[ContourPlot[x + y, Element[{x, y}, Disk[]]], 
 Graphics[{Red, PointSize[Large], Point[{x, y} /. Last[%]]}], ImageSize->300]

(*VB[*)(FrontEndRef["d9bb0245-4d86-4aa6-a23b-378bf02eeb6b"])(*,*)(*"1:eJxTTMoPSmNkYGAoZgESHvk5KRCeEJBwK8rPK3HNS3GtSE0uLUlMykkNVgEKp1gmJRkYmZjqmqRYmOmaJCaa6SYaGSfpGptbJKUZGKWmJpklAQCF/RX5"*)(*]VB*)

Minimize

Minimize a univariate function:

Minimize[2 (*SpB[*)Power[x(*|*),(*|*)2](*]SpB*) - 3 x + 5, x]

{(*FB[*)((31)(*,*)/(*,*)(8))(*]FB*),{x->(*FB[*)((3)(*,*)/(*,*)(4))(*]FB*)}}

A minimization problem containing parameters:

Minimize[a (*SpB[*)Power[x(*|*),(*|*)2](*]SpB*) + b x + c, x] 

{(*TB[*)Piecewise[{{(*|*)c(*|*),(*|*)(b==0&&a==0)||(b==0&&a>0)(*|*)},{(*|*)(*FB[*)((-((*SpB[*)Power[b(*|*),(*|*)2](*]SpB*))+4 a c)(*,*)/(*,*)(4 a))(*]FB*)(*|*),(*|*)(b>0&&a>0)||(b<0&&a>0)(*|*)},{(*|*)-Infinity(*|*),(*|*)True(*|*)}}](*|*)(*1:eJxTTMoPSmNiYGAo5gUSYZmp5S6pyflFiSX5RcGcQJGAzNTk1PLM4tRMZiAPAA9oC9E=*)(*]TB*),{x->(*TB[*)Piecewise[{{(*|*)-(*FB[*)((b)(*,*)/(*,*)(2 a))(*]FB*)(*|*),(*|*)(b>0&&a>0)||(b<0&&a>0)(*|*)},{(*|*)0(*|*),(*|*)(b==0&&a==0)||(b==0&&a>0)(*|*)},{(*|*)Indeterminate(*|*),(*|*)True(*|*)}}](*|*)(*1:eJxTTMoPSmNiYGAo5gUSYZmp5S6pyflFiSX5RcGcQJGAzNTk1PLM4tRMZiAPAA9oC9E=*)(*]TB*)}}

NMinimize

Find the global minimum of an unconstrained problem:

NMinimize[(*SpB[*)Power[x(*|*),(*|*)4](*]SpB*) - 3 (*SpB[*)Power[x(*|*),(*|*)2](*]SpB*) - x, x]

{-3.5139050389347886`,{x->1.300839565947765`}}

Find the global minimum of problems with constraints:

NMinimize[{(*SpB[*)Power[x(*|*),(*|*)2](*]SpB*) - (*SpB[*)Power[(y - 1)(*|*),(*|*)2](*]SpB*), (*SpB[*)Power[x(*|*),(*|*)2](*]SpB*) + (*SpB[*)Power[y(*|*),(*|*)2](*]SpB*) <= 4}, {x, y}]

{-9.000000014992256`,{x->2.8850141663858584`*^-10,y->-2.0000000024987092`}}

Linear Solve

Solve the matrix-vector equation

{{r, s}, {t, u}}//MatrixForm

((*GB[*){{r(*|*),(*|*)s}(*||*),(*||*){t(*|*),(*|*)u}}(*]GB*))

Substitute it to LinearSolve

LinearSolve[((*GB[*){{r(*|*),(*|*)s}(*||*),(*||*){t(*|*),(*|*)u}}(*]GB*)), {y, z}]

{(*FB[*)((u y-s z)(*,*)/(*,*)(-s t+r u))(*]FB*),(*FB[*)((t y-r z)(*,*)/(*,*)(s t-r u))(*]FB*)}

Solve a rectangular matrix equation

m = ((*GB[*){{1(*|*),(*|*)5}(*||*),(*||*){2(*|*),(*|*)6}(*||*),(*||*){3(*|*),(*|*)7}(*||*),(*||*){4(*|*),(*|*)8}}(*]GB*)); b = ((*GB[*){{9}(*||*),(*||*){10}(*||*),(*||*){11}(*||*),(*||*){12}}(*]GB*));

LinearSolve[m,b]

{{-1},{2}}

Verify

m.% == b

True

RSolve

Solve a difference equation

RSolve[a[n + 1] - 2 a[n] == 1, a[n], n]

{{a[n]->-1+(*SpB[*)Power[2(*|*),(*|*)n](*]SpB*)+((*SpB[*)Power[2(*|*),(*|*)-1+n](*]SpB*)) (*SbB[*)Subscript[C(*|*),(*|*)1](*]SbB*)}}

Include a boundary condition:

RSolve[{a[n + 1] - 2 a[n] == 1, a[0] == 1}, a[n], n]

{{a[n]->-1+(*SpB[*)Power[2(*|*),(*|*)1+n](*]SpB*)}}

And many more!

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