6.7.12 5.1

6.7.12.1 [1664] Problem 1
6.7.12.2 [1665] Problem 2
6.7.12.3 [1666] Problem 3
6.7.12.4 [1667] Problem 4
6.7.12.5 [1668] Problem 5

6.7.12.1 [1664] Problem 1

problem number 1664

Added June 26, 2019.

Problem Chapter 7.5.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a w_y + b w_z = c \ln ^k(\lambda x)+s \]

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[lambda*x]^k+s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x)+\frac {c \log ^k(\lambda x) (-\log (\lambda x))^{-k} \text {Gamma}(k+1,-\log (\lambda x))}{\lambda }+s x\right \}\right \}\]

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(lambda*x)^k+s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int \!c \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -ax+y,-bx+z \right ) \] Contains unresolve integral because maple can not integrate \(\ln ^n(x)\)

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6.7.12.2 [1665] Problem 2

problem number 1665

Added June 26, 2019.

Problem Chapter 7.5.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b w_y + c \ln (\beta y) \ln (\gamma z) w_z = k \ln (\alpha x) \]

Mathematica

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

Failed

Maple

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

\[w \left ( x,y,z \right ) ={\frac {1}{a} \left ( {\it \_F1} \left ( {\frac {-ya+bx}{b}},{\frac {-8\,b\Ei \left ( 1,3\,\ln \left ( 2 \right ) -\ln \left ( z \right ) \right ) -cy \left ( \ln \left ( \beta \,y \right ) -1 \right ) }{c}} \right ) a+xk \left ( \ln \left ( \alpha \,x \right ) -1 \right ) \right ) }\] Contains unresolve integral because maple can not integrate \(\ln ^n(x)\)

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6.7.12.3 [1666] Problem 3

problem number 1666

Added June 26, 2019.

Problem Chapter 7.5.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a \ln ^n(\beta x) w_y + b \ln ^k(\lambda x) w_z = c \ln ^m(\gamma x)+s \]

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*Log[gamma*x]+s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a (-\log (\beta x))^{-n} \log ^n(\beta x) \text {Gamma}(n+1,-\log (\beta x))}{\beta },z-\frac {b (-\log (\lambda x))^{-k} \log ^k(\lambda x) \text {Gamma}(k+1,-\log (\lambda x))}{\lambda }\right )+x (c \log (\gamma x)-c+s)\right \}\right \}\]

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*ln(gamma*x)+s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \ln \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+z \right ) -3\,\ln \left ( 2 \right ) xc+\ln \left ( x \right ) xc+ \left ( -c+s \right ) x\] Contains unresolve integral because maple can not integrate \(\ln ^n(x)\)

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6.7.12.4 [1667] Problem 4

problem number 1667

Added June 26, 2019.

Problem Chapter 7.5.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

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

Mathematica

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

\[w \left ( x,y,z \right ) =\int ^{x}\!c \left ( -3\,\ln \left ( 2 \right ) +\ln \left ( a\int \! \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{n}\,{\rm d}{\it \_f}-\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) ^{k}+s \left ( \ln \left ( \mu \, \left ( b\int \! \left ( \ln \left ( \beta \, \left ( a\int \! \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{n}\,{\rm d}{\it \_f}-\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}\,{\rm d}{\it \_f}-\int ^{x}\!b \left ( \ln \left ( \beta \, \left ( a\int \! \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}-\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) \right ) \right ) ^{r}{d{\it \_f}}+{\it \_F1} \left ( -\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!b \left ( \ln \left ( \beta \, \left ( a\int \! \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}-\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) \]

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6.7.12.5 [1668] Problem 5

problem number 1668

Added June 26, 2019.

Problem Chapter 7.5.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a_1 \ln ^{n_1}(\lambda _1 x) w_x + b_1 \ln ^{m_1}(\beta _1 y) w_y + c_1 \ln ^{k_1}(\gamma _1 z) w_z = a_2 \ln ^{n_2}(\lambda _2 x)+ b_2 \ln ^{m_2}(\beta _2 y) + c_2 \ln ^{k_2}(\gamma _2 z) \]

Mathematica

ClearAll["Global`*"]; 
pde =  a1*Log[lambda1*x]^n1*D[w[x, y,z], x] + b1*Log[beta1*y]^m1*D[w[x, y,z], y] +  c1*Log[gamma1*z]^k1*D[w[x,y,z],z]== a2*Log[lambda2*x]^n2*D[w[x, y,z], x] + b2*Log[beta2*y]^m2+  c2*Log[gamma2*z]^k2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

Failed

Maple

restart; 
local gamma; 
pde :=  a1*ln(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*ln(beta1*y)^m1*diff(w(x,y,z),y)+ c1*ln(gamma1*z)^k1*diff(w(x,y,z),z)=a2*ln(lambda2*x)^n2+ b2*ln(beta2*y)^m2+ c2*ln(gamma2*z)^k2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int ^{x}\!{\frac { \left ( \ln \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}}{{\it a1}} \left ( {\it a2}\, \left ( \ln \left ( \lambda 2\,{\it \_f} \right ) \right ) ^{{\it n2}}+{\it b2}\, \left ( \ln \left ( \beta 2\,\RootOf \left ( \int \! \left ( \ln \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \ln \left ( \beta 1\,{\it \_a} \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}{d{\it \_a}}-\int \! \left ( \ln \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y \right ) \right ) \right ) ^{{\it m2}}+{\it c2}\, \left ( \ln \left ( \gamma 2\,\RootOf \left ( \int \! \left ( \ln \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \ln \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \ln \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it k2}} \right ) }{d{\it \_f}}+{\it \_F1} \left ( -\int \! \left ( \ln \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y,-\int \! \left ( \ln \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}\,{\rm d}z \right ) \] Contains RootOf and unresolved integrals \(\ln ^n(x)\)

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