x

y

%y

%y

ymin = -2.025210907987559
ymax = 2.2819881313212087
yscale: ymax - ymin
xmax = 1.4759411329067151
xmin = -2.7601484883002563
xscale: xmax - xmin
a22: 4
a21: 3
a12: 2
a11: 1
discrim: (a11 - a22)^(2) + ((4 * a12) * a21)
lambda2: (a11 + a22 - sqrt(discrim)) / 2
u2version2: a11 ≟ lambda2 ∧ a12 ≟ 0
mag2 = If(u2version2, sqrt((lambda2 - a22)^(2) + a21^(2)), sqrt(a12^(2) + (lambda2 - a11)^(2)))
u2: Vector[If[u2version2, (lambda2 - a22, a21), (a12, lambda2 - a11)]] / mag2
lambda1: (a11 + a22 + sqrt(discrim)) / 2
u1version2: a11 ≟ lambda1 ∧ a12 ≟ 0
mag1 = If(u1version2, sqrt((lambda1 - a22)^(2) + a21^(2)), sqrt(a12^(2) + (lambda1 - a11)^(2)))
u1: Vector[If[u1version2, (lambda1 - a22, a21), (a12, lambda1 - a11)]] / mag1
LL = Corner(1)
UR = Corner(3)
xlabeldisplayed: "" + (LaTeX[xlabel]) + ""
ylabeldisplayed: "" + (LaTeX[ylabel]) + ""
Aequation: "\begin{align*}A = \begin{bmatrix} " + a11 + " & " + a12 + "\\ " + a21 + "&" + a22 + "\end{bmatrix}\end{align*}"
lambdaComplexReal: (a11 + a22) / 2
lambdaComplexImaginary: sqrt((-discrim)) / 2
A = (0.9476997412933758, -0.7249244055547234)
a = CurveCartesian((ℯ^((lambdaComplexReal * t)) * ((x(A) * cos((lambdaComplexImaginary * t))) - ((1 / lambdaComplexImaginary * ((x(A) * (lambdaComplexReal - a11)) - (a12 * y(A)))) * sin((lambdaComplexImaginary * t))))), (ℯ^((lambdaComplexReal * t)) * ((y(A) * cos((lambdaComplexImaginary * t))) - ((1 / lambdaComplexImaginary * ((x(A) * (lambdaComplexImaginary^(2) + (lambdaComplexReal - a11)^(2)) / a12) - (y(A) * (lambdaComplexReal - a11)))) * sin((lambdaComplexImaginary * t))))), t, 0, 10)
lambda1Complex = If(discrim ≥ 0, (a11 + a22 + sqrt(discrim)) / 2 + (0 * ί), (a11 + a22) / 2 + (sqrt((-discrim)) / 2 * ί))
lambda2Complex = If(discrim ≥ 0, (a11 + a22 - sqrt(discrim)) / 2 + (0 * ί), (a11 + a22) / 2 - (sqrt((-discrim)) / 2 * ί))
u11complex = If(u1version2, lambda1 - a22 + (0 * ί), a12 + (0 * ί))
u12complex = If(u1version2, a21 + (0 * ί), lambda1Complex - a11)
u21complex = If(u2version2, lambda2 - a22 + (0 * ί), a12 + (0 * ί))
u22complex = If(u2version2, a21 + (0 * ί), lambda2Complex - a11)
c1: ((x(A) * u22complex) - (y(A) * u21complex)) / ((u11complex * u22complex) - (u12complex * u21complex))
c2: ((y(A) * u11complex) - (x(A) * u12complex)) / ((u11complex * u22complex) - (u12complex * u21complex))
identityMultiple: a12 ≟ 0 ∧ a21 ≟ 0 ∧ a22 ≟ a11
v11 = If(u1version2, If[a21 ≟ 0, 0, y(u1) / a21], If[a11 ≟ lambda1, 0, x(u1) / (a11 - lambda1)])
v12 = If(u1version2, If[a21 ≟ 0, y(u1) / (a22 - lambda1), 0], If[a11 ≟ lambda1, x(u1) / a12, 0])
cd1: ((x(A) * v12) - (y(A) * v11)) / ((x(u1) * v12) - (v11 * y(u1)))
cd2: ((y(A) * x(u1)) - (x(A) * y(u1))) / ((x(u1) * v12) - (v11 * y(u1)))
c11real = If(identityMultiple, x(A), real((c1 * u11complex)))
c21real = If(identityMultiple, 0, real((c2 * u21complex)))
c12real = If(identityMultiple, y(A), real((c1 * u12complex)))
c22real = If(identityMultiple, 0, real((c2 * u22complex)))
c11complex: x(A)
c21complex: (-1 / lambdaComplexImaginary * ((x(A) * (lambdaComplexReal - a11)) - (a12 * y(A))))
c12complex: y(A)
c22complex: (-1 / lambdaComplexImaginary * ((x(A) * (lambdaComplexImaginary^(2) + (lambdaComplexReal - a11)^(2)) / a12) - (y(A) * (lambdaComplexReal - a11))))
lambda1t = If(lambda1 ≟ 1, "t", If[lambda1 ≟ "-1", "-t", lambda1 + "t"])
lambda2t = If(lambda2 ≟ 1, "t", If[lambda2 ≟ "-1", "-t", lambda2 + "t"])
xrealtext: If[c11real ≠ 0, c11real + If[lambda1 ≟ 0, "", "e^{" + lambda1t + "}"] + (If[c21real > 0, "+", ""]), ""] + (If[c21real ≠ 0, c21real + If[lambda2 ≠ 0, "e^{" + lambda2t + "}", ""], ""])
yrealtext: If[c12real ≠ 0, c12real + If[lambda1 ≟ 0, "", "e^{" + lambda1t + "}"] + (If[c22real > 0, "+", ""]), ""] + (If[c22real ≠ 0, c22real + If[lambda2 ≠ 0, "e^{" + lambda2t + "}", ""], ""])
c11degen = If(identityMultiple, x(A), (cd1 * x(u1)) + (cd2 * v11))
c12degen = If(identityMultiple, y(A), (cd1 * y(u1)) + (cd2 * v12))
lambdaComplexRealt = If(lambdaComplexReal ≟ 1, "t", If[lambdaComplexReal ≟ -1, "-t", lambdaComplexReal + "t"])
lambdaComplexImaginaryt = If(lambdaComplexImaginary ≟ 1, "t", If[lambdaComplexImaginary ≟ -1, "-t", lambdaComplexImaginary + "t"])
xcomplextext: If[c11complex ≠ 0, c11complex + "\cos(" + lambdaComplexImaginaryt + ")" + (If[lambdaComplexReal ≟ 0, "", "e^{" + lambdaComplexRealt + "}"]) + (If[c21complex > 0, "+", ""]), ""] + (If[c21complex ≠ 0, c21complex + "\sin(" + lambdaComplexImaginaryt + ")" + (If[lambdaComplexReal ≟ 0, "", "e^{" + lambdaComplexRealt + "}"])])
ycomplextext: If[c12complex ≠ 0, c12complex + "\cos(" + lambdaComplexImaginaryt + ")" + (If[lambdaComplexReal ≟ 0, "", "e^{" + lambdaComplexRealt + "}"]) + (If[c22complex > 0, "+", ""]), ""] + (If[c22complex ≠ 0, c22complex + "\sin(" + lambdaComplexImaginaryt + ")" + (If[lambdaComplexReal ≟ 0, "", "e^{" + lambdaComplexRealt + "}"])])
c22degen = If(identityMultiple, 0, (cd2 * y(u1)))
c21degen = If(identityMultiple, 0, (cd2 * x(u1)))
xdegentext: If[c11degen ≠ 0, c11degen + If[lambda1 ≟ 0, "", "e^{" + lambda1t + "}"] + (If[c21degen > 0, "+", ""]), ""] + (If[c21degen ≠ 0, c21degen + "t" + (If[lambda1 ≠ 0, "e^{" + lambda1t + "}", ""]), ""])
ydegentext: If[c12degen ≠ 0, c12degen + If[lambda1 ≟ 0, "", "e^{" + lambda1t + "}"] + (If[c22degen > 0, "+", ""]), ""] + (If[c22degen ≠ 0, c22degen + "t" + (If[lambda1 ≠ 0, "e^{" + lambda1t + "}", ""]), ""])
solutiontext: "\begin{align*}x(t) &= " + (If[lambda1 ≟ lambda2, xdegentext, If[discrim > 0, xrealtext, xcomplextext]]) + "\\y(t) &= " + (If[lambda1 ≟ lambda2, ydegentext, If[discrim > 0, yrealtext, ycomplextext]]) + "\end{align*}"
scale: sqrt(xscale^(2) + yscale^(2))
u1p1: (u1 * scale / 5)
u1p2: ((-u1) * scale / 5)
u1n1 = Vector(((x(u1), y(u1)) * scale / 5), (0, 0))
u1n2 = Vector((((-x(u1)), (-y(u1))) * scale / 5), (0, 0))
u1line: (x * y(u1)) = (y * x(u1))
u2line: (x * y(u2)) = (y * x(u2))
u2p1: (u2 * scale / 5)
u2p2: ((-u2) * scale / 5)
u2n1 = Vector(((x(u2), y(u2)) * scale / 5), (0, 0))
u2n2 = Vector((((-x(u2)), (-y(u2))) * scale / 5), (0, 0))
x0 = 0.9476997412933758
y0 = -0.7249244055547234
systemtext: "\begin{align*}\diff{x}{t} &=" + (If[a11 ≠ 0, a11 + "x" + (If[a12 > 0, "+", ""]), ""]) + "" + (If[a12 ≠ 0, a12 + "y", ""]) + "" + (If[a11 ≟ 0 ∧ a12 ≟ 0, "0", ""]) + "\\ \diff{y}{t} &=" + (If[a21 ≠ 0, a21 + "x" + (If[a22 > 0, "+", ""]), ""]) + "" + (If[a22 ≠ 0, a22 + "y", ""]) + "" + (If[a21 ≟ 0 ∧ a22 ≟ 0, "0", ""]) + "\\ x(0) &= " + x0 + "\\ y(0) &= " + y0 + "\end{align*}"
hmm: real(u11complex)
eigenvectorEquation: "\begin{align*}\vc{u}_1 &= \begin{bmatrix}" + (If[discrim ≥ 0, real(u11complex) + "\\", u11complex + "\\"]) + (If[discrim ≥ 0, real(u12complex) + "", u12complex + ""]) + "\end{bmatrix}\\ \vc{u}_2 &= \begin{bmatrix}" + (If[discrim ≥ 0, real(u21complex) + "\\", u21complex + "\\"]) + (If[discrim ≥ 0, real(u22complex) + "", u22complex + ""]) + "\end{bmatrix}\end{align*}"
lambdaEquation: "\lambda = " + (LaTeX[If[discrim > 0, lambda1 + ("," * lambda2), lambdaComplexReal + "\pm" + lambdaComplexImaginary + "ί "]]) + ""
tmax: 10
b = CurveCartesian(real(((c1 * u11complex) * ℯ^((lambda1Complex * t))) + ((c2 * u21complex) * ℯ^((lambda2Complex * t)))), real(((c1 * u12complex) * ℯ^((lambda1Complex * t))) + ((c2 * u22complex) * ℯ^((lambda2Complex * t)))), t, 0, tmax)
c = CurveCartesian((x(A) * ℯ^((lambda1 * t))), (y(A) * ℯ^((lambda1 * t))), t, 0, tmax)
d = CurveCartesian((ℯ^((lambda1 * t)) * ((cd1 * x(u1)) + (cd2 * ((t * x(u1)) + v11)))), (ℯ^((lambda1 * t)) * ((cd1 * y(u1)) + (cd2 * ((t * y(u1)) + v12)))), t, 0, tmax)
tmin: (-0.1 * tmax)
xymax = Max(xmax, ymax)
xymin = Min(xmin, ymin)
Ax = Point(yAxis)
Ay = Point(yAxis)
bx(t) = If[0 ≤ t ≤ tmax, real(((c1 * u11complex) * ℯ^((lambda1Complex * t))) + ((c2 * u21complex) * ℯ^((lambda2Complex * t))))]
by(t) = If[0 ≤ t ≤ tmax, real(((c1 * u12complex) * ℯ^((lambda1Complex * t))) + ((c2 * u22complex) * ℯ^((lambda2Complex * t))))]
cx(t) = If[0 ≤ t ≤ tmax, (x(A) * ℯ^((lambda1 * t)))]
cy(t) = If[0 ≤ t ≤ tmax, (y(A) * ℯ^((lambda1 * t)))]
dx(t) = If[0 ≤ t ≤ tmax, (ℯ^((lambda1 * t)) * ((cd1 * x(u1)) + (cd2 * ((t * x(u1)) + v11))))]
dy(t) = If[0 ≤ t ≤ tmax, (ℯ^((lambda1 * t)) * ((cd1 * y(u1)) + (cd2 * ((t * y(u1)) + v12))))]