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'));
 

\[w \left (x , y , z\right ) = \left (k \int \ln \left (\lambda x \right )^{m} {\mathrm e}^{-c \int \ln \left (\beta x \right )^{n}d x}d x +f_{1} \left (-a x +y , -b x +z \right )\right ) {\mathrm e}^{c \int \ln \left (\beta x \right )^{n}d x}\]

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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'));
 

\[w \left (x , y , z\right ) = \left (s \int \ln \left (\mu x \right )^{m} {\mathrm e}^{-c x}d x +f_{1} \left (-a \int \ln \left (\beta x \right )^{n}d x +y , -b \int \ln \left (\lambda x \right )^{k}d x +z \right )\right ) {\mathrm e}^{c x}\]

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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]];
 

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

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'));
 

\[w \left (x , y , z\right ) = \left (s \int \ln \left (\mu x \right )^{m} {\mathrm e}^{-a x}d x +f_{1} \left (-b \int \ln \left (\beta x \right )^{n}d x +y , -c \int _{}^{x}{\ln \left (\lambda \left (b \int \ln \left (\beta \textit {\_b} \right )^{n}d \textit {\_b} -b \int \ln \left (\beta x \right )^{n}d x +y \right )\right )}^{k}d \textit {\_b} +z \right )\right ) {\mathrm e}^{a x}\]

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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'));
 

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

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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]];
 

\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^x\frac {a \log ^{-\text {n1}}(\text {lambda1} K[3])}{\text {b1}}dK[3]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[4]}\frac {a \log ^{-\text {n1}}(\text {lambda1} K[3])}{\text {b1}}dK[3]\right ) \left (\text {c1} \log ^{\text {k1}}(\text {beta1} K[4])+\text {c2} \log ^{\text {k2}}(\text {beta2} K[4])+\text {c3} \log ^{\text {k3}}(\text {beta3} K[4])\right ) \log ^{-\text {n1}}(\text {lambda1} K[4])}{\text {b1}}dK[4]+c_1\left (y-\int _1^x\frac {\text {b2} \log ^{-\text {n1}}(\text {lambda1} K[1]) \log ^{\text {n2}}(\text {lambda2} K[1])}{\text {b1}}dK[1],z-\int _1^x\frac {\text {b3} \log ^{-\text {n1}}(\text {lambda1} K[2]) \log ^{\text {n3}}(\text {lambda3} K[2])}{\text {b1}}dK[2]\right )\right )\right \}\right \}\]

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'));
 

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