Added July 1, 2019.
Problem Chapter 8.3.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a e^{\lambda x} w_y + b e^{\beta x} w_z = c e^{\gamma x} w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Exp[lambda*x]*D[w[x, y,z], y] +b*Exp[beta*x]*D[w[x,y,z],z]== c*Exp[gamma*x]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c e^{\gamma x}}{\gamma }} c_1\left (y-\frac {a e^{\lambda x}}{\lambda },z-\frac {b e^{\beta x}}{\beta }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*exp(lambda*x)*diff(w(x,y,z),y)+b*exp(beta*x)*diff(w(x,y,z),z)= c*exp(gamma*x)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {\lambda \,y-a{{\rm e}^{x\lambda }}}{\lambda }},{\frac {\beta \,z-b{{\rm e}^{\beta \,x}}}{\beta }} \right ) {{\rm e}^{{\frac {c{{\rm e}^{x\gamma }}}{\gamma }}}}\]
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Added July 1, 2019.
Problem Chapter 8.3.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a e^{\lambda x} w_y + b e^{\beta x} w_z = (c e^{\gamma y}+s e^{\mu z}) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Exp[lambda*x]*D[w[x, y,z], y] +b*Exp[beta*x]*D[w[x,y,z],z]== (c*Exp[gamma*y]+s Exp[mu*z])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a e^{\lambda x}}{\lambda },z-\frac {b e^{\beta x}}{\beta }\right ) \exp \left (\frac {c \text {Ei}\left (\frac {a e^{\lambda x} \gamma }{\lambda }\right ) e^{\gamma \left (y-\frac {a e^{\lambda x}}{\lambda }\right )}}{\lambda }+\frac {s \text {Ei}\left (\frac {b e^{\beta x} \mu }{\beta }\right ) e^{\mu \left (z-\frac {b e^{\beta x}}{\beta }\right )}}{\beta }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*exp(lambda*x)*diff(w(x,y,z),y)+b*exp(beta*x)*diff(w(x,y,z),z)= (c*exp(gamma*y)+s*exp(mu*z))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {\lambda \,y-a{{\rm e}^{x\lambda }}}{\lambda }},{\frac {\beta \,z-b{{\rm e}^{\beta \,x}}}{\beta }} \right ) {{\rm e}^{{\frac {1}{\lambda \,\beta } \left ( -s{{\rm e}^{{\frac {\mu \, \left ( \beta \,z-b{{\rm e}^{\beta \,x}} \right ) }{\beta }}}}\Ei \left ( 1,-{\frac {\mu \,b{{\rm e}^{\beta \,x}}}{\beta }} \right ) \lambda -c{{\rm e}^{{\frac {\gamma \, \left ( \lambda \,y-a{{\rm e}^{x\lambda }} \right ) }{\lambda }}}}\Ei \left ( 1,-{\frac {\gamma \,a{{\rm e}^{x\lambda }}}{\lambda }} \right ) \beta \right ) }}}\]
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Added July 2, 2019.
Problem Chapter 8.3.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a e^{\lambda y} w_y + b e^{\beta y} w_z = (c e^{\gamma x}+s e^{\mu z}) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Exp[lambda*y]*D[w[x, y,z], y] +b*Exp[beta*y]*D[w[x,y,z],z]== (c*Exp[gamma*x]+s Exp[mu*z])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {a \lambda x+e^{-\lambda y}}{\lambda },\frac {b \left (e^{-\lambda y}\right )^{1-\frac {\beta }{\lambda }}}{a (\lambda -\beta )}+z\right ) \exp \left (\int _1^x\left (e^{\gamma K[1]} c+\exp \left (-\frac {\mu \left (b \lambda (x-K[1]) \left (a \lambda (x-K[1])+e^{-\lambda y}\right )^{-\frac {\beta }{\lambda }}+(\beta -\lambda ) z+\frac {b e^{-\lambda y} \left (\left (a \lambda (x-K[1])+e^{-\lambda y}\right )^{-\frac {\beta }{\lambda }}-\left (e^{-\lambda y}\right )^{-\frac {\beta }{\lambda }}\right )}{a}\right )}{\lambda -\beta }\right ) s\right )dK[1]\right )\right \}\right \}\] Generates Solve::incnst: Inconsistent or redundant transcendental equation
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*exp(lambda*y)*diff(w(x,y,z),y)+b*exp(beta*y)*diff(w(x,y,z),z)= (c*exp(gamma*x)+s*exp(mu*z))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-x\lambda \,a-{{\rm e}^{-\lambda \,y}}}{a\lambda }},{\frac {1}{a \left ( \beta -\lambda \right ) } \left ( -b{{\rm e}^{-\lambda \,y}} \left ( \left ( {{\rm e}^{-\lambda \,y}} \right ) ^{-1} \right ) ^{{\frac {\beta }{\lambda }}}+az \left ( \beta -\lambda \right ) \right ) } \right ) {{\rm e}^{\int ^{x}\!c{{\rm e}^{{\it \_a}\,\gamma }}+s{{\rm e}^{{\frac {\mu }{a \left ( \beta -\lambda \right ) } \left ( -b{{\rm e}^{-\lambda \,y}} \left ( \left ( {{\rm e}^{-\lambda \,y}} \right ) ^{-1} \right ) ^{{\frac {\beta }{\lambda }}}+b \left ( {{\rm e}^{-\lambda \,y}}+a\lambda \, \left ( x-{\it \_a} \right ) \right ) \left ( \left ( {{\rm e}^{-\lambda \,y}}+a\lambda \, \left ( x-{\it \_a} \right ) \right ) ^{-1} \right ) ^{{\frac {\beta }{\lambda }}}+az \left ( \beta -\lambda \right ) \right ) }}}{d{\it \_a}}}}\]
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Added July 2, 2019.
Problem Chapter 8.3.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (A_1 e^{\alpha _1 x} + B_1 e^{\nu _1 x+\lambda y}) w_y + (A_2 e^{\alpha _2 x} + B_2 e^{\nu _2 x+\beta z}) w_z = k e^{\gamma z} w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (A1*Exp[alpha1*x] + B1*Exp[nu1*x+lambda*y])*D[w[x, y,z], y] + (A2*Exp[alpha2*x] + B2*Exp[nu2*x+beta*z])*D[w[x,y,z],z]== k*Exp[gamma*x]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+(A1*exp(alpha1*x) + B1*exp(nu1*x+lambda*y))*diff(w(x,y,z),y)+(A2*exp(alpha2*x) + B2*exp(nu2*x+beta*z))*diff(w(x,y,z),z)= k*exp(gamma*x)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{\lambda } \left ( -{\it B1}\,\int \!{{\rm e}^{\nu 1\,x+{\frac {\lambda \,{\it A1}\,{{\rm e}^{\alpha 1\,x}}}{\alpha 1}}}}\,{\rm d}x\lambda -{{\rm e}^{{\frac {\lambda \, \left ( {\it A1}\,{{\rm e}^{\alpha 1\,x}}-\alpha 1\,y \right ) }{\alpha 1}}}} \right ) },{\frac {1}{\beta } \left ( -{\it B2}\,\int \!{{\rm e}^{\nu 2\,x+{\frac {\beta \,{\it A2}\,{{\rm e}^{\alpha 2\,x}}}{\alpha 2}}}}\,{\rm d}x\beta -{{\rm e}^{{\frac {\beta \, \left ( {\it A2}\,{{\rm e}^{\alpha 2\,x}}-\alpha 2\,z \right ) }{\alpha 2}}}} \right ) } \right ) {{\rm e}^{{\frac {k{{\rm e}^{x\gamma }}}{\gamma }}}}\]
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Added July 2, 2019.
Problem Chapter 8.3.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a e^{\alpha x} w_x + b e^{\beta y} w_y + c e^{\gamma z} w_z = k e^{\lambda x} w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[alpha*x]*D[w[x, y,z], x] + b*Exp[beta*y]*D[w[x, y,z], y] + c*Exp[gamma*z]*D[w[x,y,z],z]== k*Exp[lambda*x]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {k e^{x (\lambda -\alpha )}}{a (\lambda -\alpha )}} c_1\left (\frac {b e^{-\alpha x}}{a \alpha }-\frac {e^{-\beta y}}{\beta },\frac {c e^{-\alpha x}}{a \alpha }-\frac {e^{-\gamma z}}{\gamma }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*exp(alpha*x)*diff(w(x,y,z),x)+b*exp(beta*y)*diff(w(x,y,z),y)+c*exp(gamma*z)*diff(w(x,y,z),z)= k*exp(lambda*x)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac { \left ( {{\rm e}^{\beta \,y}}\beta \,b-a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\beta \,y}}}{\alpha \,b\beta }},{\frac { \left ( {{\rm e}^{\gamma \,z}}c\gamma -a\alpha \,{{\rm e}^{\alpha \,x}} \right ) {{\rm e}^{-\alpha \,x-\gamma \,z}}}{\alpha \,c\gamma }} \right ) {{\rm e}^{-{\frac {k{{\rm e}^{-x \left ( \alpha -\lambda \right ) }}}{ \left ( \alpha -\lambda \right ) a}}}}\]
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Added July 2, 2019.
Problem Chapter 8.3.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a e^{\beta y} w_x + b e^{\alpha x} w_y + c e^{\gamma z} w_z = k e^{\lambda x} w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*Exp[beta*y]*D[w[x, y,z], x] + b*Exp[alpha*x]*D[w[x, y,z], y] + c*Exp[gamma*z]*D[w[x,y,z],z]== k*Exp[lambda*x]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a*exp(beta*y)*diff(w(x,y,z),x)+b*exp(alpha*x)*diff(w(x,y,z),y)+c*exp(gamma*z)*diff(w(x,y,z),z)= k*exp(lambda*x)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-{{\rm e}^{\alpha \,x}}b\beta +{{\rm e}^{\beta \,y}}a\alpha }{\alpha \,b\beta }},-{\frac {\beta \,b}{ \left ( -{{\rm e}^{\alpha \,x}}b\beta +{{\rm e}^{\beta \,y}}a\alpha \right ) \alpha \,c\gamma } \left ( a{{\rm e}^{\beta \,y}}{{\rm e}^{-\gamma \,z}}\alpha -{{\rm e}^{\alpha \,x}}{{\rm e}^{-\gamma \,z}}b\beta +\alpha \,c\gamma \,x-\ln \left ( {\frac {{{\rm e}^{\beta \,y}}a\alpha }{\beta \,b}} \right ) \gamma \,c \right ) } \right ) {{\rm e}^{\int ^{x}\!{\frac {{{\rm e}^{{\it \_a}\,\lambda }}k\alpha }{{{\rm e}^{\beta \,y}}a\alpha -b\beta \, \left ( {{\rm e}^{\alpha \,x}}-{{\rm e}^{\alpha \,{\it \_a}}} \right ) }}{d{\it \_a}}}}\]
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Added July 2, 2019.
Problem Chapter 8.3.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ (a_1+a_2 e^{\alpha x}) w_x + (b_1+b_2 e^{\beta y}) w_y + (c_1+c_2 e^{\gamma z}) w_z = (k_1+k_2 e^{\alpha x}) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = (a1+a2*Exp[alpha*x])*D[w[x, y,z], x] + (b1+b2*Exp[beta*y])*D[w[x, y,z], y] + (c1+c2*Exp[gamma*z])*D[w[x,y,z],z]== (k1+k2*Exp[alpha*x])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {\text {k1} x}{\text {a1}}} \left (\text {a1}+\text {a2} e^{\alpha x}\right )^{\frac {\text {a1} \text {k2}-\text {a2} \text {k1}}{\text {a1} \text {a2} \alpha }} c_1\left (\frac {\log \left (\frac {e^{\beta y} \left (\text {a1}+\text {a2} e^{\alpha x}\right )^{\frac {\text {b1} \beta }{\text {a1} \alpha }}}{\text {b1}+\text {b2} e^{\beta y}}\right )}{\text {b1} \beta }-\frac {x}{\text {a1}},\frac {\log \left (\frac {e^{\gamma z} \left (\text {a1}+\text {a2} e^{\alpha x}\right )^{\frac {\text {c1} \gamma }{\text {a1} \alpha }}}{\text {c1}+\text {c2} e^{\gamma z}}\right )}{\text {c1} \gamma }-\frac {x}{\text {a1}}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := (a1+a2*exp(alpha*x))*diff(w(x,y,z),x)+(b1+b2*exp(beta*y))*diff(w(x,y,z),y)+(c1+c2*exp(gamma*z))*diff(w(x,y,z),z)= (k1+k2*exp(alpha*x))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{\alpha \,{\it a1}\,\beta \,{\it b1}} \left ( -\alpha \,{\it a1}\,\RootOf \left ( y\alpha \,{\it a1}\,\beta -\alpha \,{\it a1}\,\ln \left ( {\frac {-{\it b1}+{{\rm e}^{{\it \_Z}}}}{{\it b2}} \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) ^{{\frac {\beta \,{\it b1}}{\alpha \,{\it a1}}}}} \right ) +\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) \beta \,{\it b1} \right ) -\beta \, \left ( -\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) {\it b1}+\alpha \, \left ( -y{\it a1}+{\it b1}\,x \right ) \right ) \right ) },{\frac {1}{\alpha \,{\it a1}\,\gamma \,{\it c1}} \left ( -\alpha \,{\it a1}\,\RootOf \left ( z\alpha \,{\it a1}\,\gamma -\alpha \,{\it a1}\,\ln \left ( {\frac {-{\it c1}+{{\rm e}^{{\it \_Z}}}}{{\it c2}} \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) ^{{\frac {\gamma \,{\it c1}}{\alpha \,{\it a1}}}}} \right ) +\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) {\it c1}\,\gamma \right ) -\gamma \, \left ( -{\it c1}\,\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) +\alpha \, \left ( -z{\it a1}+{\it c1}\,x \right ) \right ) \right ) } \right ) \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) ^{{\frac {{\it k2}}{\alpha \,{\it a2}}}-{\frac {{\it k1}}{\alpha \,{\it a1}}}} \left ( {{\rm e}^{\alpha \,x}} \right ) ^{{\frac {{\it k1}}{\alpha \,{\it a1}}}}\]
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Added July 2, 2019.
Problem Chapter 8.3.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ e^{\beta y} (a_1+a_2 e^{\alpha x}) w_x + e^{\alpha x} (b_1+b_2 e^{\beta y}) w_y + c e^{\beta y+\gamma z} w_z = k_3 e^{\beta y} (k_1+k_2 e^{\alpha x}) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = Exp[beta*y]*(a1+a2*Exp[alpha*x])*D[w[x, y,z], x] + Exp[alpha*x]*(b1+b2*Exp[beta*y])*D[w[x, y,z], y] + c*Exp[beta*y+gamma*z]*D[w[x,y,z],z]== k3*Exp[beta*y]*(k1+k2*Exp[alpha*x])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {\text {k3} \left ((\text {a1} \text {k2}-\text {a2} \text {k1}) \log \left (\text {a1}+\text {a2} e^{\alpha x}\right )+\text {a2} \alpha \text {k1} x\right )}{\text {a1} \text {a2} \alpha }\right ) c_1\left (\frac {c \log \left (\text {a1}+\text {a2} e^{\alpha x}\right )}{\text {a1} \alpha }-\frac {c x}{\text {a1}}-\frac {e^{-\gamma z}}{\gamma },\frac {\log \left (\text {b1}+\text {b2} e^{\beta y}\right )}{\text {b2} \beta }-\frac {\log \left (\text {a1}+\text {a2} e^{\alpha x}\right )}{\text {a2} \alpha }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := exp(beta*y)*(a1+a2*exp(alpha*x))*diff(w(x,y,z),x)+exp(alpha*x)*(b1+b2*exp(beta*y))*diff(w(x,y,z),y)+c*exp(beta*y+gamma*z)*diff(w(x,y,z),z)= k3*exp(beta*y)*(k1+k2*exp(alpha*x))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{\alpha \,{\it a2}\,\beta \,{\it b2}} \left ( y\alpha \,{\it a2}\,\beta -\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) \beta \,{\it b2}+\alpha \,{\it a2}\,\RootOf \left ( y\alpha \,{\it a2}\,\beta -\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) \beta \,{\it b2}-\alpha \,{\it a2}\,\ln \left ( {\frac {{\it b1}}{-{\it b2}+{{\rm e}^{{\it \_Z}}}} \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) ^{-{\frac {\beta \,{\it b2}}{\alpha \,{\it a2}}}}} \right ) \right ) \right ) },{\frac {\ln \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) \gamma \,c-\alpha \, \left ( c\gamma \,x+{{\rm e}^{-\gamma \,z}}{\it a1} \right ) }{{\it a1}\,\alpha \,c\gamma }} \right ) \left ( {\it a1}+{\it a2}\,{{\rm e}^{\alpha \,x}} \right ) ^{{\frac {{\it k3}\,{\it k2}}{\alpha \,{\it a2}}}-{\frac {{\it k3}\,{\it k1}}{\alpha \,{\it a1}}}} \left ( {{\rm e}^{\alpha \,x}} \right ) ^{{\frac {{\it k3}\,{\it k1}}{\alpha \,{\it a1}}}}\]
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