6.9.12 5.1

6.9.12.1 [1996] Problem 1

problem number 1996

Added Jan 19, 2020.

Problem Chapter 9.5.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \ln ^n(\beta x) w + k \ln ^m(\lambda x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+ a*D[w[x,y,z],y]+b*D[w[x,y,z],z]==c*Log[beta*x]^n*w[x,y,z]+ k*Log[lambda*x]^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^xc \log ^n(\beta K[1])dK[1]\right ) \left (\int _1^x\exp \left (-\int _1^{K[2]}c \log ^n(\beta K[1])dK[1]\right ) k \log ^m(\lambda K[2])dK[2]+c_1(y-a x,z-b x)\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+a*diff(w(x,y,z),y)+ b*diff(w(x,y,z),z)=c*ln(beta*x)^n*w(x,y,z)+ k*ln(lambda*x)^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) = \left ( \int \!k \left ( \ln \left ( \lambda \,x \right ) \right ) ^{m}{{\rm e}^{-c\int \! \left ( \ln \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x}}\,{\rm d}x+{\it \_F1} \left ( -ax+y,-bx+z \right ) \right ) {{\rm e}^{\int \! \left ( \ln \left ( \beta \,x \right ) \right ) ^{n}c\,{\rm d}x}}\]

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6.9.12.2 [1997] Problem 2

problem number 1997

Added Jan 19, 2020.

Problem Chapter 9.5.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + a \ln ^n(\beta x)w_y + b \ln ^k(\lambda x) w_z = c w + s \ln ^m(\mu x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+ a*Log[beta*x]^n*D[w[x,y,z],y]+b*Log[lambda*x]^k*D[w[x,y,z],z]==c*w[x,y,z]+ s*Log[mu*x]^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{c x} \left (\int _1^xe^{-c K[3]} s \log ^m(\mu K[3])dK[3]+c_1\left (y-\int _1^xa \log ^n(\beta K[1])dK[1],z-\int _1^xb \log ^k(\lambda K[2])dK[2]\right )\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+a*ln(beta*x)^n*diff(w(x,y,z),y)+ b*ln(lambda*x)^k*diff(w(x,y,z),z)=c*w(x,y,z)+ s*ln(mu*x)^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) = \left ( \int \!s \left ( \ln \left ( \mu \,x \right ) \right ) ^{m}{{\rm e}^{-xc}}\,{\rm d}x+{\it \_F1} \left ( -\int \!a \left ( \ln \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+z \right ) \right ) {{\rm e}^{xc}}\]

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6.9.12.3 [1998] Problem 3

problem number 1998

Added Jan 19, 2020.

Problem Chapter 9.5.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + b \ln ^n(\beta x)w_y + c \ln ^k(\lambda y) w_z = a w + s \ln ^m(\mu x) \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+ b*Log[beta*x]^n*D[w[x,y,z],y]+c*Log[lambda*y]^k*D[w[x,y,z],z]==a*w[x,y,z]+ s*Log[mu*x]^m; 
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)+b*ln(beta*x)^n*diff(w(x,y,z),y)+ c*ln(lambda*y)^k*diff(w(x,y,z),z)=a*w(x,y,z)+ s*ln(mu*x)^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) = \left ( \int \!s \left ( \ln \left ( \mu \,x \right ) \right ) ^{m}{{\rm e}^{-ax}}\,{\rm d}x+{\it \_F1} \left ( -\int \!b \left ( \ln \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!c \left ( \ln \left ( \lambda \, \left ( b\int \! \left ( \ln \left ( \beta \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b}-\int \!b \left ( \ln \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{k}{d{\it \_b}}+z \right ) \right ) {{\rm e}^{ax}}\]

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6.9.12.4 [1999] Problem 4

problem number 1999

Added Jan 19, 2020.

Problem Chapter 9.5.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ a \ln (\alpha x) w_x + b \ln (\beta y) w_y + c \ln (\gamma z) w_z = p w + q \ln (\lambda x) \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*Log[alpha*x]*D[w[x,y,z],x]+ b*Log[beta*y]*D[w[x,y,z],y]+c*Log[gamma*z]*D[w[x,y,z],z]==p*w[x,y,z]+ q*Log[lambda*x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde := a*ln(alpha*x)*diff(w(x,y,z),x)+b*ln(beta*y)*diff(w(x,y,z),y)+ c*ln(gamma*z)*diff(w(x,y,z),z)=p*w(x,y,z)+ q*ln(lambda*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) = \left ( \int \!{\frac {q\ln \left ( \lambda \,x \right ) }{a\ln \left ( \alpha \,x \right ) }{{\rm e}^{{\frac {p\Ei \left ( 1,-\ln \left ( \alpha \,x \right ) \right ) }{a\alpha }}}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {\Ei \left ( 1,-\ln \left ( \alpha \,x \right ) \right ) b\beta -a\Ei \left ( 1,-\ln \left ( \beta \,y \right ) \right ) \alpha }{\alpha \,b\beta }},{\frac {\Ei \left ( 1,-\ln \left ( \alpha \,x \right ) \right ) c\gamma -a\Ei \left ( 1,-\ln \left ( \gamma \,z \right ) \right ) \alpha }{\alpha \,c\gamma }} \right ) \right ) {{\rm e}^{-{\frac {p\Ei \left ( 1,-\ln \left ( \alpha \,x \right ) \right ) }{a\alpha }}}}\]

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6.9.12.5 [2000] Problem 5

problem number 2000

Added Jan 19, 2020.

Problem Chapter 9.5.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ b_1 \ln ^{n_1}(\lambda _1 x) w_x + b_2 \ln ^{n_2}(\lambda _2 y) w_y + b_3 \ln ^{n_3}(\lambda _3 z) w_z = a w + c_1 \ln ^{k_1}(\beta _1 x)+ c_2 \ln ^{k_2}(\beta _2 y)+ c_3 \ln ^{k_3}(\beta _3 z) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  b1*Log[lambda1*x]^n1*D[w[x,y,z],x]+ b2*Log[lambda2*x]^n2*D[w[x,y,z],y]+b3*Log[lambda3*x]^n3*D[w[x,y,z],z]==a*w[x,y,z]+ c1*Log[beta1*x]^k1+ c2*Log[beta2*x]^k2+ c3*Log[beta3*x]^k3; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {a (-\log (\text {lambda1} x))^{\text {n1}} \log ^{-\text {n1}}(\text {lambda1} x) \text {Gamma}(1-\text {n1},-\log (\text {lambda1} x))}{\text {b1} \text {lambda1}}\right ) \left (\int _1^x\frac {\exp \left (-\frac {a \text {Gamma}(1-\text {n1},-\log (\text {lambda1} K[3])) (-\log (\text {lambda1} K[3]))^{\text {n1}} \log ^{-\text {n1}}(\text {lambda1} K[3])}{\text {b1} \text {lambda1}}\right ) \left (\text {c1} \log ^{\text {k1}}(\text {beta1} K[3])+\text {c2} \log ^{\text {k2}}(\text {beta2} K[3])+\text {c3} \log ^{\text {k3}}(\text {beta3} K[3])\right ) \log ^{-\text {n1}}(\text {lambda1} K[3])}{\text {b1}}dK[3]+c_1\left (y-\int _1^x\frac {\text {b2} \log ^{-\text {n1}}(\text {lambda1} K[1]) \log ^{\text {n2}}(\text {lambda2} K[1])}{\text {b1}}dK[1],z-\int _1^x\frac {\text {b3} \log ^{-\text {n1}}(\text {lambda1} K[2]) \log ^{\text {n3}}(\text {lambda3} K[2])}{\text {b1}}dK[2]\right )\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := b__1*ln(lambda__1*x)^(n__1)*diff(w(x,y,z),x)+b__2*ln(lambda__2*x)^(n__2)*diff(w(x,y,z),y)+ b__3*ln(lambda__3*x)^(n__3)*diff(w(x,y,z),z)=a*w(x,y,z)+ c__1*ln(beta__1*x)^(k__1)+ c__2*ln(beta__2*x)^(k__2)+ c__3*ln(beta__3*x)^(k__3); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) = \left ( \int \!{\frac { \left ( c_{1}\, \left ( \ln \left ( \beta _{1}\,x \right ) \right ) ^{k_{1}}+c_{2}\, \left ( \ln \left ( \beta _{2}\,x \right ) \right ) ^{k_{2}}+c_{3}\, \left ( \ln \left ( \beta _{3}\,x \right ) \right ) ^{k_{3}} \right ) \left ( \ln \left ( \lambda _{1}\,x \right ) \right ) ^{-n_{1}}}{b_{1}}{{\rm e}^{-{\frac {a\int \! \left ( \ln \left ( \lambda _{1}\,x \right ) \right ) ^{-n_{1}}\,{\rm d}x}{b_{1}}}}}}\,{\rm d}x+{\it \_F1} \left ( -{\frac {b_{2}\,\int \! \left ( \ln \left ( \lambda _{2}\,x \right ) \right ) ^{n_{2}} \left ( \ln \left ( \lambda _{1}\,x \right ) \right ) ^{-n_{1}}\,{\rm d}x}{b_{1}}}+y,-{\frac {b_{3}\,\int \! \left ( \ln \left ( \lambda _{3}\,x \right ) \right ) ^{n_{3}} \left ( \ln \left ( \lambda _{1}\,x \right ) \right ) ^{-n_{1}}\,{\rm d}x}{b_{1}}}+z \right ) \right ) {{\rm e}^{\int \!{\frac {a \left ( \ln \left ( \lambda _{1}\,x \right ) \right ) ^{-n_{1}}}{b_{1}}}\,{\rm d}x}}\]