Algebraic and Analytic Methods - 1.13 Differential Equations

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
1.13.E2	${\displaystyle{\displaystyle w(z)=Aw_{1}(z)+Bw_{2}(z)}}$ w(z) = Aw_{1}(z)+Bw_{2}(z)		w(z) = Aw[1](z)+ Bw[2](z)	w[z] == Subscript[Aw, 1][z]+ Subscript[Bw, 2][z]	Skipped - no semantic math	Skipped - no semantic math	-	-
1.13#Ex1	${\displaystyle{\displaystyle w_{1}(z_{0})=1}}$ w_{1}(z_{0}) = 1		w[1](z[0]) = 1	Subscript[w, 1][Subscript[z, 0]] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
1.13#Ex3	${\displaystyle{\displaystyle w_{2}(z_{0})=0}}$ w_{2}(z_{0}) = 0		w[2](z[0]) = 0	Subscript[w, 2][Subscript[z, 0]] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
1.13.E6	${\displaystyle{\displaystyle Aw_{1}(z)+Bw_{2}(z)=0}}$ Aw_{1}(z)+Bw_{2}(z) = 0		Aw[1](z)+ Bw[2](z) = 0	Subscript[Aw, 1][z]+ Subscript[Bw, 2][z] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
1.13.E9	${\displaystyle{\displaystyle w(z)=w_{0}(z)+Aw_{1}(z)+Bw_{2}(z)}}$ w(z) = w_{0}(z)+Aw_{1}(z)+Bw_{2}(z)		w(z) = w[0](z)+ Aw[1](z)+ Bw[2](z)	w[z] == Subscript[w, 0][z]+ Subscript[Aw, 1][z]+ Subscript[Bw, 2][z]	Skipped - no semantic math	Skipped - no semantic math	-	-
1.13.E10	${\displaystyle{\displaystyle w_{0}(z)=w_{2}(z)\int\frac{w_{1}(z)r(z)}{\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}}\mathrm{d}z-w_{1}(z)\int\frac{w_{2}(z)r(z)}{\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}}\mathrm{d}z}}$ w_{0}(z) = w_{2}(z)\int\frac{w_{1}(z)r(z)}{\Wronskian@{w_{1}(z),w_{2}(z)}}\diff{z}-w_{1}(z)\int\frac{w_{2}(z)r(z)}{\Wronskian@{w_{1}(z),w_{2}(z)}}\diff{z}		w[0](z) = w[2](z)* int((w[1](z)* r(z))/((w[1](z))*diff(w[2](z), z)-diff(w[1](z), z)*(w[2](z))), z)- w[1](z)* int((w[2](z)* r(z))/((w[1](z))*diff(w[2](z), z)-diff(w[1](z), z)*(w[2](z))), z)	Subscript[w, 0][z] == Subscript[w, 2][z]* Integrate[Divide[Subscript[w, 1][z]* r[z],Wronskian[{Subscript[w, 1][z], Subscript[w, 2][z]}, z]], z, GenerateConditions->None]- Subscript[w, 1][z]* Integrate[Divide[Subscript[w, 2][z]* r[z],Wronskian[{Subscript[w, 1][z], Subscript[w, 2][z]}, z]], z, GenerateConditions->None]	Translation Error	Translation Error	-	-
1.13#Ex5	${\displaystyle{\displaystyle W(\xi)=w\left(\frac{1}{\xi}\right)}}$ W(\xi) = w\left(\frac{1}{\xi}\right)		W(xi) = w((1)/(xi))	W[\[Xi]] == w[Divide[1,\[Xi]]]	Skipped - no semantic math	Skipped - no semantic math	-	-
1.13#Ex8	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}U}{{\mathrm{d}z}^{2}}+IU=0}}$ \deriv[2]{U}{z}+IU = 0		diff(U, [z$(2)])+ I*U = 0	D[U, {z, 2}]+ I*U == 0	Failure	Failure	Failed [300 / 300] Result: .5000000004+.8660254040*I Test Values: {I = 1/2*3^(1/2)+1/2*I, U = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: .5000000004+.8660254040*I Test Values: {I = 1/2*3^(1/2)+1/2*I, U = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} Result: .5000000004+.8660254040*I Test Values: {I = 1/2*3^(1/2)+1/2*I, U = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)} Result: .5000000004+.8660254040*I Test Values: {I = 1/2*3^(1/2)+1/2*I, U = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [210 / 210] Result: Complex[0.8660254037844387, 0.49999999999999994] Test Values: {Rule[Complex[0, 1], 1], Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.7320508075688774, 0.9999999999999999] Test Values: {Rule[Complex[0, 1], 2], Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
1.13#Ex9	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}V}{{\mathrm{d}z}^{2}}+JV=0}}$ \deriv[2]{V}{z}+JV = 0		diff(V, [z$(2)])+ J*V = 0	D[V, {z, 2}]+ J*V == 0	Failure	Failure	Failed [300 / 300] Result: .5000000004+.8660254040*I Test Values: {J = 1/2*3^(1/2)+1/2*I, V = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: .5000000004+.8660254040*I Test Values: {J = 1/2*3^(1/2)+1/2*I, V = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} Result: .5000000004+.8660254040*I Test Values: {J = 1/2*3^(1/2)+1/2*I, V = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)} Result: .5000000004+.8660254040*I Test Values: {J = 1/2*3^(1/2)+1/2*I, V = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.5000000000000001, 0.8660254037844386] Test Values: {Rule[J, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.5000000000000001, 0.8660254037844386] Test Values: {Rule[J, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data