6.6.13 5.2

6.6.13.1 [1498] Problem 1

problem number 1498

Added May 26, 2019.

Problem Chapter 6.5.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a x^n w_y + b \ln ^k(\lambda x) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*x^n*D[w[x, y,z], y] +b*Log[lambda*x]^k*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+ a*x^n*diff(w(x,y,z),y)+b*ln(lambda*x)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {-a x x^{n}+\left (n +1\right ) y}{n +1}, -b x \ln \left (\lambda x \right )+b x +z \right )\]

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6.6.13.2 [1499] Problem 2

problem number 1499

Added May 26, 2019.

Problem Chapter 6.5.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y + c \ln ^k(\lambda x)) w_y + (b z+ s \ln ^n(\lambda x)) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a*y+c*Log[lambda*x]^k)*D[w[x, y,z], y] +(b*z+s*Log[lambda*x]^n)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y e^{-a x}-\int _1^xc e^{-a K[1]} (\log (\lambda )+\log (K[1]))^kdK[1],z e^{-b x}-\int _1^xe^{-b K[2]} s (\log (\lambda )+\log (K[2]))^ndK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+ (a*y+c*ln(lambda*x)^k)*diff(w(x,y,z),y)+(b*z+s*log(lambda*x)^n)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.6.13.3 [1500] Problem 3

problem number 1500

Added May 26, 2019.

Problem Chapter 6.5.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y + (c \ln ^n(\lambda x)+ s \ln ^k(\beta y) ) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x*D[w[x, y,z], x] + b*y*D[w[x, y,z], y] +(c*Log[lambda*x]^n+s*Log[beta*y]^k)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*x*diff(w(x,y,z),x)+  b*y*diff(w(x,y,z),y)+(c*ln(lambda*x)^n+s*ln(beta*y)^k)*diff(w(x,y,z),z)= 0; 
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 x^{-\frac {b}{a}}, \frac {\left (n +1\right ) \left (i \pi \,\mathrm {csgn}\left (i \beta \right ) \mathrm {csgn}\left (i y \right ) \mathrm {csgn}\left (i \beta y \right )+i \pi \mathrm {csgn}\left (i y \right )^{3}-i \pi \mathrm {csgn}\left (i y \right )^{2} \mathrm {csgn}\left (i x^{\frac {b}{a}}\right )+i \pi \mathrm {csgn}\left (i \beta y \right )^{3}-i \left (\mathrm {csgn}\left (i y \right )-\mathrm {csgn}\left (i x^{\frac {b}{a}}\right )\right ) \pi \,\mathrm {csgn}\left (i y \right ) \mathrm {csgn}\left (i y x^{-\frac {b}{a}}\right )+\left (-i \pi \,\mathrm {csgn}\left (i \beta \right )-i \pi \,\mathrm {csgn}\left (i y \right )\right ) \mathrm {csgn}\left (i \beta y \right )^{2}-2 \ln \left (\beta \right )-2 \ln \left (x^{\frac {b}{a}}\right )-2 \ln \left (y x^{-\frac {b}{a}}\right )\right ) a s \left (-\frac {i \pi \,\mathrm {csgn}\left (i \beta \right ) \mathrm {csgn}\left (i y \right ) \mathrm {csgn}\left (i \beta y \right )}{2}-\frac {i \pi \mathrm {csgn}\left (i y \right )^{3}}{2}+\frac {i \pi \mathrm {csgn}\left (i y \right )^{2} \mathrm {csgn}\left (i x^{\frac {b}{a}}\right )}{2}-\frac {i \pi \mathrm {csgn}\left (i \beta y \right )^{3}}{2}+\frac {i \left (\mathrm {csgn}\left (i y \right )-\mathrm {csgn}\left (i x^{\frac {b}{a}}\right )\right ) \pi \,\mathrm {csgn}\left (i y \right ) \mathrm {csgn}\left (i y x^{-\frac {b}{a}}\right )}{2}+\frac {\left (i \pi \,\mathrm {csgn}\left (i \beta \right )+i \pi \,\mathrm {csgn}\left (i y \right )\right ) \mathrm {csgn}\left (i \beta y \right )^{2}}{2}+\ln \left (\beta \right )+\ln \left (x^{\frac {b}{a}}\right )+\ln \left (y x^{-\frac {b}{a}}\right )\right )^{k}+\left (2 \left (n +1\right ) a z +\left (i \pi \,\mathrm {csgn}\left (i \lambda \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \lambda x \right )+i \pi \mathrm {csgn}\left (i \lambda x \right )^{3}-i \left (\mathrm {csgn}\left (i \lambda \right )+\mathrm {csgn}\left (i x \right )\right ) \pi \mathrm {csgn}\left (i \lambda x \right )^{2}-2 \ln \left (\lambda \right )-2 \ln \left (x \right )\right ) c \left (-\frac {i \pi \,\mathrm {csgn}\left (i \lambda \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \lambda x \right )}{2}-\frac {i \pi \mathrm {csgn}\left (i \lambda x \right )^{3}}{2}+\frac {i \left (\mathrm {csgn}\left (i \lambda \right )+\mathrm {csgn}\left (i x \right )\right ) \pi \mathrm {csgn}\left (i \lambda x \right )^{2}}{2}+\ln \left (\lambda \right )+\ln \left (x \right )\right )^{n}\right ) \left (k +1\right ) b}{2 \left (n +1\right ) \left (k +1\right ) a b}\right )\]

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6.6.13.4 [1501] Problem 4

problem number 1501

Added May 26, 2019.

Problem Chapter 6.5.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x \ln (\lambda x) w_x + b y \ln (\beta y) w_y + c z \ln (\gamma z)w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x*Log[lambda*x]*D[w[x, y,z], x] + b*y*Log[beta*y]*D[w[x, y,z], y] +c*Log[gamma*z]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*x*ln(lambda*x)*diff(w(x,y,z),x)+ b*y*ln(beta*y)*diff(w(x,y,z),y)+c*ln(gamma*z)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {-a \ln \left (\ln \left (\beta y \right )\right )+b \ln \left (\ln \left (\lambda x \right )\right )}{a}, \frac {-b \Ei \left (1, -\ln \left (z \right )-\ln \left (\gamma \right )\right )-\gamma c \ln \left (\ln \left (\beta y \right )\right )}{\gamma c}\right )\]

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6.6.13.5 [1502] Problem 5

problem number 1502

Added May 26, 2019.

Problem Chapter 6.5.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x \ln (\lambda x) w_x + b y \ln (\beta y) w_y + c z \ln (\gamma x)w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x*Log[lambda*x]*D[w[x, y,z], x] + b*y*Log[beta*y]*D[w[x, y,z], y] +c*Log[gamma*x]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*x*ln(lambda*x)*diff(w(x,y,z),x)+ b*y*ln(beta*y)*diff(w(x,y,z),y)+c*ln(gamma*x)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a \ln \left (\ln \left (\beta y \right )\right )-b \ln \left (\ln \left (\lambda x \right )\right )}{b}, \frac {2 a z -2 c \ln \left (x \right )-\left (i \pi \,\mathrm {csgn}\left (i \lambda \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \lambda x \right )+i \pi \mathrm {csgn}\left (i \lambda x \right )^{3}-i \left (\mathrm {csgn}\left (i \lambda \right )+\mathrm {csgn}\left (i x \right )\right ) \pi \mathrm {csgn}\left (i \lambda x \right )^{2}-2 \ln \left (\lambda \right )+2 \ln \left (\gamma \right )\right ) c \ln \left (-\frac {i \pi \,\mathrm {csgn}\left (i \lambda \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \lambda x \right )}{2}-\frac {i \pi \mathrm {csgn}\left (i \lambda x \right )^{3}}{2}+\frac {i \left (\mathrm {csgn}\left (i \lambda \right )+\mathrm {csgn}\left (i x \right )\right ) \pi \mathrm {csgn}\left (i \lambda x \right )^{2}}{2}+\ln \left (\lambda \right )+\ln \left (x \right )\right )}{2 a}\right )\]

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6.6.13.6 [1503] Problem 6

problem number 1503

Added May 26, 2019.

Problem Chapter 6.5.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x \ln ^n(x) w_x + b y \ln ^m(y) w_y + c z \ln ^k(z)w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x*Log[x]^n*D[w[x, y,z], x] + b*y*Log[y]^m*D[w[x, y,z], y] +c*z*Log[z]^k*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b \log ^{1-n}(x)}{a (n-1)}-(m-1)^{\frac {1}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m,\frac {c \log ^{1-n}(x)}{a (n-1)}-(k-1)^{\frac {1}{k-1}} \log (z) \left (\frac {(k-1)^{\frac {1}{1-k}}}{\log (z)}\right )^k\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*x*ln(x)^n*diff(w(x,y,z),x)+ b*y*ln(y)^m*diff(w(x,y,z),y)+c*z*ln(z)^k*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {-\left (n -1\right ) a \ln \left (y \right )^{-m +1}+\left (m -1\right ) b \ln \left (x \right )^{-n +1}}{\left (n -1\right ) \left (m -1\right ) b}, \frac {-\left (n -1\right ) a \ln \left (z \right )^{-k +1}+\left (k -1\right ) c \ln \left (x \right )^{-n +1}}{\left (n -1\right ) \left (k -1\right ) c}\right )\]

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