Numerical Methods - 3.7 Ordinary Differential Equations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
3.7#Ex10 A 11 ⁢ ( τ , z ) = ∑ s = 0 ∞ τ s s ! ⁢ f s ⁢ ( z ) subscript 𝐴 11 𝜏 𝑧 superscript subscript 𝑠 0 superscript 𝜏 𝑠 𝑠 subscript 𝑓 𝑠 𝑧 {\displaystyle{\displaystyle A_{11}(\tau,z)=\sum_{s=0}^{\infty}\frac{\tau^{s}}% {s!}f_{s}(z)}}
A_{11}(\tau,z) = \sum_{s=0}^{\infty}\frac{\tau^{s}}{s!}f_{s}(z)		 

A[11](tau , z) = sum(((tau)^(s))/(factorial(s))*f[s](z), s = 0..infinity)
 
Subscript[A, 11][\[Tau], z] == Sum[Divide[\[Tau]^(s),(s)!]*Subscript[f, s][z], {s, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
3.7#Ex11 A 12 ⁢ ( τ , z ) = ∑ s = 0 ∞ τ s s ! ⁢ g s ⁢ ( z ) subscript 𝐴 12 𝜏 𝑧 superscript subscript 𝑠 0 superscript 𝜏 𝑠 𝑠 subscript 𝑔 𝑠 𝑧 {\displaystyle{\displaystyle A_{12}(\tau,z)=\sum_{s=0}^{\infty}\frac{\tau^{s}}% {s!}g_{s}(z)}}
A_{12}(\tau,z) = \sum_{s=0}^{\infty}\frac{\tau^{s}}{s!}g_{s}(z)		 

A[12](tau , z) = sum(((tau)^(s))/(factorial(s))*g[s](z), s = 0..infinity)
 
Subscript[A, 12][\[Tau], z] == Sum[Divide[\[Tau]^(s),(s)!]*Subscript[g, s][z], {s, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
3.7#Ex12 A 21 ⁢ ( τ , z ) = ∑ s = 0 ∞ τ s s ! ⁢ f s + 1 ⁢ ( z ) subscript 𝐴 21 𝜏 𝑧 superscript subscript 𝑠 0 superscript 𝜏 𝑠 𝑠 subscript 𝑓 𝑠 1 𝑧 {\displaystyle{\displaystyle A_{21}(\tau,z)=\sum_{s=0}^{\infty}\frac{\tau^{s}}% {s!}f_{s+1}(z)}}
A_{21}(\tau,z) = \sum_{s=0}^{\infty}\frac{\tau^{s}}{s!}f_{s+1}(z)		 

A[21](tau , z) = sum(((tau)^(s))/(factorial(s))*f[s + 1](z), s = 0..infinity)
 
Subscript[A, 21][\[Tau], z] == Sum[Divide[\[Tau]^(s),(s)!]*Subscript[f, s + 1][z], {s, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
3.7#Ex13 A 22 ⁢ ( τ , z ) = ∑ s = 0 ∞ τ s s ! ⁢ g s + 1 ⁢ ( z ) subscript 𝐴 22 𝜏 𝑧 superscript subscript 𝑠 0 superscript 𝜏 𝑠 𝑠 subscript 𝑔 𝑠 1 𝑧 {\displaystyle{\displaystyle A_{22}(\tau,z)=\sum_{s=0}^{\infty}\frac{\tau^{s}}% {s!}g_{s+1}(z)}}
A_{22}(\tau,z) = \sum_{s=0}^{\infty}\frac{\tau^{s}}{s!}g_{s+1}(z)		 

A[22](tau , z) = sum(((tau)^(s))/(factorial(s))*g[s + 1](z), s = 0..infinity)
 
Subscript[A, 22][\[Tau], z] == Sum[Divide[\[Tau]^(s),(s)!]*Subscript[g, s + 1][z], {s, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
3.7#Ex14 b 1 ⁢ ( τ , z ) = ∑ s = 0 ∞ τ s s ! ⁢ h s ⁢ ( z ) subscript 𝑏 1 𝜏 𝑧 superscript subscript 𝑠 0 superscript 𝜏 𝑠 𝑠 subscript ℎ 𝑠 𝑧 {\displaystyle{\displaystyle b_{1}(\tau,z)=\sum_{s=0}^{\infty}\frac{\tau^{s}}{% s!}h_{s}(z)}}
b_{1}(\tau,z) = \sum_{s=0}^{\infty}\frac{\tau^{s}}{s!}h_{s}(z)		 

b[1](tau , z) = sum(((tau)^(s))/(factorial(s))*h[s](z), s = 0..infinity)
 
Subscript[b, 1][\[Tau], z] == Sum[Divide[\[Tau]^(s),(s)!]*Subscript[h, s][z], {s, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
3.7#Ex15 b 2 ⁢ ( τ , z ) = ∑ s = 0 ∞ τ s s ! ⁢ h s + 1 ⁢ ( z ) subscript 𝑏 2 𝜏 𝑧 superscript subscript 𝑠 0 superscript 𝜏 𝑠 𝑠 subscript ℎ 𝑠 1 𝑧 {\displaystyle{\displaystyle b_{2}(\tau,z)=\sum_{s=0}^{\infty}\frac{\tau^{s}}{% s!}h_{s+1}(z)}}
b_{2}(\tau,z) = \sum_{s=0}^{\infty}\frac{\tau^{s}}{s!}h_{s+1}(z)		 

b[2](tau , z) = sum(((tau)^(s))/(factorial(s))*h[s + 1](z), s = 0..infinity)
 
Subscript[b, 2][\[Tau], z] == Sum[Divide[\[Tau]^(s),(s)!]*Subscript[h, s + 1][z], {s, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
3.7.E13 𝐀𝐰 = 𝐛 𝐀𝐰 𝐛 {\displaystyle{\displaystyle\mathbf{A}\mathbf{w}=\mathbf{b}}}
\mathbf{A}\mathbf{w} = \mathbf{b}		 

A*w = b
 
A*w == b
 Skipped - no semantic math Skipped - no semantic math - -
3.7.E15 d 2 w k d x 2 + ( λ k - q ⁢ ( x ) ) ⁢ w k = 0 derivative subscript 𝑤 𝑘 𝑥 2 subscript 𝜆 𝑘 𝑞 𝑥 subscript 𝑤 𝑘 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w_{k}}{{\mathrm{d}x}^{2}}+(% \lambda_{k}-q(x))w_{k}=0}}
\deriv[2]{w_{k}}{x}+(\lambda_{k}-q(x))w_{k} = 0		 

diff(w[k], [x$(2)])+(lambda[k]- q(x))*w[k] = 0

D[Subscript[w, k], {x, 2}]+(Subscript[\[Lambda], k]- q[x])*Subscript[w, k] == 0
Failure Failure
Failed [300 / 300]
Result: -.2500000002-.4330127020*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, x = 1.5, lambda[k] = 1/2*3^(1/2)+1/2*I, w[k] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.2500000002-.4330127020*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, x = 1.5, lambda[k] = 1/2*3^(1/2)+1/2*I, w[k] = 1/2*3^(1/2)+1/2*I, k = 2}


Result: -.2500000002-.4330127020*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, x = 1.5, lambda[k] = 1/2*3^(1/2)+1/2*I, w[k] = 1/2*3^(1/2)+1/2*I, k = 3}


Result: .4330127020-.2500000002*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, x = 1.5, lambda[k] = 1/2*3^(1/2)+1/2*I, w[k] = -1/2+1/2*I*3^(1/2), k = 1}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.25000000000000006, -0.43301270189221924]
Test Values: {Rule[k, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[λ, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.25000000000000006, -0.43301270189221924]
Test Values: {Rule[k, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[λ, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.7.E16 w k ⁢ ( a ) = w k ⁢ ( b ) subscript 𝑤 𝑘 𝑎 subscript 𝑤 𝑘 𝑏 {\displaystyle{\displaystyle w_{k}(a)=w_{k}(b)}}
w_{k}(a) = w_{k}(b)		 

w[k](a) = w[k](b)
 
Subscript[w, k][a] == Subscript[w, k][b]
 Skipped - no semantic math Skipped - no semantic math - -
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