Wronskian
Wronskian[{y1,y2,…},x]
gives the Wronskian determinant for the functions y1,y2,… depending on x.
Wronskian[eqn,y,x]
gives the Wronskian determinant for the basis of the solutions of the linear differential equation eqn with dependent variable y and independent variable x.
Wronskian[eqns,{y1,y2,…},x]
gives the Wronskian determinant for the system of linear differential equations eqns.
Examples
open all close allBasic Examples (3)
These functions are linearly independent:
Wronskian[{Exp[x], Exp[2x]}, x]Wronskian[{Sin[x], Cos[x], x Sin[x], x Cos[x]}, x]These functions are dependent:
Wronskian[{Exp[x], Exp[x + 3]}, x]Wronskian[{x ^ 2, 3x ^ 2 + 5x + 1, x ^ 2 + 5x + 1}, x]The Wronskian for a linear equation:
Wronskian[y''[x] - x y[x] == 0, y, x]Except for a constant, the result is the same as for the explicit solution:
DSolve[y''[x] - x y[x] == 0, y, x]Wronskian[{AiryAi[x], AiryBi[x]}, x]Scope (11)
Functions (6)
Wronskian[{x, x ^ 2}, x]Wronskian[{x ^ n, x ^ m}, x]The last element can be expressed as a linear combination of the previous ones:
Wronskian[{1, x, x ^ 2, (x - 3)(x - 4)}, x]Reduce[ForAll[x, a + b x + c x ^ 2 == (x - 3)(x - 4)], {a, b, c}]Wronskian[{(1/x), (1/x + 1)}, x]Wronskian[{(1/x), (1/x + 1), (1/x(x + 1))}, x]Exponentials and exponential polynomials:
Wronskian[{3 ^ x, x 3 ^ x}, x]Wronskian[{3 ^ x, 3 ^ (x + 1)}, x]Wronskian[{Cos[x], Sin[x]}, x]Wronskian[{Cos[x], Sin[x], Sin[x + Pi / 3]}, x]Wronskian[{Cos[x], Cos[2x]}, x]Wronskian[{Cos[x], x Cos[x]}, x]Wronskian[{ChebyshevT[1, x], ChebyshevT[2, x]}, x]Wronskian[{ChebyshevT[0, x], ChebyshevT[1, x], (x - 4)}, x]Wronskian[{BesselJ[1, x], BesselJ[2, x]}, x]Plot[%, {x, 0, 10}]Wronskian[{AiryAi[x], AiryBi[x]}, x]Differential Equations (5)
Constant coefficient linear equation:
Wronskian[y'''[x] - 5y''[x] + 11y[x] == 0, y, x]The Wronskian for a differential equation is usually simpler than for its solution:
DSolve[y'''[x] - 5y''[x] + 11y[x] == 0, y, x]Wronskian[Coefficient[y[x] /. %[[1]], Table[C[i], {i, 3}]], x]Polynomial coefficient linear equation:
Wronskian[y''[x] - x y'[x] + y[x] == 0, y, x]The corresponding Wronskian from the general solution:
DSolve[y''[x] - x y'[x] + y[x] == 0, y, x]Wronskian[Coefficient[y[x] /. %[[1]], {C[1], C[2]}], x]Special function coefficients:
Wronskian[y''[x] + Sinc[x] y[x] == 0, y, x]Wronskian[y''[x] + y'[x] + BesselJ[1, x] y[x] == 0, y, x]Wronskian[y'''[x] + HypergeometricPFQ[{a1, a2}, {b1}, x] y[x] == 0, y, x]System of first-order differential equations:
Wronskian[{y'[x] == y[x] - z[x], z'[x] == y[x] + z[x]}, {y, z}, x]System with a higher-order equation:
eqns = {y''[x] = z[x], z'[x] == y[x] - z[x]};Compute the Wronskian by conversion to a system of first-order ODEs:
feqns = {y'[x] == v[x], v'[x] == z[x], z'[x] == y[x] - z[x]};Wronskian[feqns, {v, y, z}, x]Applications (2)
Variation of parameters formula for forced second-order differential equations:
VariationOfParameters[homogeneouseqn_, forcingterm_, y_, x_] :=
Block[{sol, y1, y2, w, v1, v2},
sol = DSolve[homogeneouseqn, y, x];
y1 = y[x] /. sol[[1]] /. {C[1] -> 1, C[2] -> 0};
y2 = y[x] /. sol[[1]] /. {C[1] -> 0, C[2] -> 1};
w = Wronskian[{y1, y2}, x];
v1 = -Integrate[y2 forcingterm / w, x];
v2 = Integrate[y1 forcingterm / w, x];
Simplify[y1 v1 + y2 v2]
]VariationOfParameters[y''[x] - 4 y[x] == 0, x ^ 2, y, x]Verify that the components of the general solution for an ODE are linearly independent:
DSolve[x y'''[x] + x y'[x] - y[x] == 0, y, x]Wronskian[Coefficient[y[x] /. %[[1]], Table[C[i], {i, 3}]], x]Properties & Relations (5)
Wronskian is equivalent to a determinant:
Wronskian[{f[x], g[x]}, x]Det[Table[D[{f[x], g[x]}, {x, m}], {m, 0, 1}]]Wronskian detects linear dependence:
Wronskian[{f[x], c f[x]}, x]Wronskian[{f[x] + g[x], a f[x] + b g[x]}, x]Casoratian performs linear dependence for sequences of a discrete argument:
Casoratian[{2 ^ n, n 2 ^ n}, n]Use Orthogonalize to generate a set of linearly independent functions:
basis = Orthogonalize[{1, x, x ^ 2}, Integrate[#1 #2, {x, -1, 1}]&]Express a function in terms of the basis:
coef = Integrate[(x - 1)(x - 2)basis, {x, -1, 1}]{coef.basis, (x - 1)(x - 2)}//ExpandThe last component is linearly dependent on the previous ones:
Orthogonalize[{1, x, x ^ 2, (x - 1)(x - 2)}, Integrate[#1 #2, {x, -1, 1}]&]Use Reduce to express polynomials and rational functions in terms of each other:
Reduce[ForAll[x, a + b x + c x ^ 2 == (x - 3)(x - 4)], {a, b}]Reduce[ForAll[x, x ≠ 0 && x ≠ 1, (a/x) + (b/x - 1) == (1/x(x - 1))], {a, b}]Neat Examples (1)
The differential equation for Kelvin functions:
Wronskian[x ^ 4y''''[x] + 2x ^ 3y'''[x] - (1 + 2 v ^ 2)(x ^ 2y''[x] - x y'[x]) + (v ^ 4 - 4v ^ 2 + x ^ 4)y[x] == 0, y, x]Compare to the general solution:
DSolve[x ^ 4y''''[x] + 2x ^ 3y'''[x] - (1 + 2 v ^ 2)(x ^ 2y''[x] - x y'[x]) + (v ^ 4 - 4v ^ 2 + x ^ 4)y[x] == 0, y, x]Wronskian[Coefficient[y[x] /. %[[1]], Table[C[i], {i, 4}]], x]//FunctionExpandText
Wolfram Research (2008), Wronskian, Wolfram Language function, https://reference.wolfram.com/language/ref/Wronskian.html.
CMS
Wolfram Language. 2008. "Wronskian." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Wronskian.html.
APA
Wolfram Language. (2008). Wronskian. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Wronskian.html