6.8.12 5.1

6.8.12.1 [1832] Problem 1

problem number 1832

Added Oct 17, 2019.

Problem Chapter 8.5.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{\int c \ln \left (\beta x \right )^{n}d x}\]

____________________________________________________________________________________

6.8.12.2 [1833] Problem 2

problem number 1833

Added Oct 17, 2019.

Problem Chapter 8.5.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, z -\left (\int \frac {c \ln \left (\beta x \right )^{n}}{a}d x \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {s \ln \left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \lambda }{a}\right )^{m}}{a}d\mathit {\_a}}\]

____________________________________________________________________________________

6.8.12.3 [1834] Problem 3

problem number 1834

Added Oct 17, 2019.

Problem Chapter 8.5.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y -\left (\int a \ln \left (\beta x \right )^{n}d x \right ), z -\left (\int b \ln \left (\lambda x \right )^{k}d x \right )\right ) {\mathrm e}^{\int c \ln \left (\gamma x \right )^{m}d x}\]

____________________________________________________________________________________

6.8.12.4 [1835] Problem 4

problem number 1835

Added Oct 17, 2019.

Problem Chapter 8.5.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

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

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y -\left (\int a \ln \left (\beta x \right )^{n}d x \right ), z -\left (\int _{}^{x}b \ln \left (\left (a \left (\int \ln \left (\mathit {\_b} \beta \right )^{n}d \mathit {\_b} \right )+y -\left (\int a \ln \left (\beta x \right )^{n}d x \right )\right ) \lambda \right )^{k}d\mathit {\_b} \right )\right ) {\mathrm e}^{\int c \ln \left (\gamma x \right )^{m}d x}\]

____________________________________________________________________________________

6.8.12.5 [1836] Problem 5

problem number 1836

Added Oct 17, 2019.

Problem Chapter 8.5.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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], x]*D[w[x, y,z], y] + c1*Log[gamma1*z]^k1*D[w[x,y,z],z]== ( a2*Log[lambda2*x]^n2+ b2*Log[beta2*y]^m2+ c2*Log[gamma2*z]^k2)*w[x,y,z]; 
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)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-\left (\int \ln \left (\lambda 1 x \right )^{-\mathit {n1}}d x \right )+\int \frac {\mathit {a1} \ln \left (\beta 1 y \right )^{-\mathit {m1}}}{\mathit {b1}}d y , -\left (\int \ln \left (\lambda 1 x \right )^{-\mathit {n1}}d x \right )+\int \frac {\mathit {a1} \ln \left (\gamma 1 z \right )^{-\mathit {k1}}}{\mathit {c1}}d z \right ) {\mathrm e}^{\int _{}^{x}\frac {\left (\mathit {a2} \ln \left (\mathit {\_f} \lambda 2 \right )^{\mathit {n2}}+\mathit {b2} \ln \left (\beta 2 \RootOf \left (\int \ln \left (\mathit {\_f} \lambda 1 \right )^{-\mathit {n1}}d \mathit {\_f} -\left (\int \ln \left (\lambda 1 x \right )^{-\mathit {n1}}d x \right )+\int \frac {\mathit {a1} \ln \left (\beta 1 y \right )^{-\mathit {m1}}}{\mathit {b1}}d y -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {a1} \ln \left (\mathit {\_a} \beta 1 \right )^{-\mathit {m1}}}{\mathit {b1}}d\mathit {\_a} \right )\right )\right )^{\mathit {m2}}+\mathit {c2} \ln \left (\gamma 2 \RootOf \left (\int \ln \left (\mathit {\_f} \lambda 1 \right )^{-\mathit {n1}}d \mathit {\_f} -\left (\int \ln \left (\lambda 1 x \right )^{-\mathit {n1}}d x \right )+\int \frac {\mathit {a1} \ln \left (\gamma 1 z \right )^{-\mathit {k1}}}{\mathit {c1}}d z -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {a1} \ln \left (\mathit {\_a} \gamma 1 \right )^{-\mathit {k1}}}{\mathit {c1}}d\mathit {\_a} \right )\right )\right )^{\mathit {k2}}\right ) \ln \left (\mathit {\_f} \lambda 1 \right )^{-\mathit {n1}}}{\mathit {a1}}d\mathit {\_f}}\]

____________________________________________________________________________________