6.8.22 7.4

6.8.22.1 [1886] Problem 1

problem number 1886

Added Nov 30, 2019.

Problem Chapter 8.7.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y + b w_z = c \arccot ^n(\beta x) w \]

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

\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\int _1^xc \cot ^{-1}(\beta K[1])^ndK[1]\right )\right \}\right \}\]

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

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

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6.8.22.2 [1887] Problem 2

problem number 1887

Added Nov 30, 2019.

Problem Chapter 8.7.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a_1 w_x + a_2 w_y + a_3 w_z = \left ( b_1 \arccot (\lambda _1 x)+b_2 \arccot (\lambda _2 y)+b_3 \arccot (\lambda _3 z) \right ) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a1*D[w[x,y,z],x]+a2*D[w[x,y,z],y]+a3*D[w[x,y,z],z]== (b1*ArcCot[lambda1*x]+b2*ArcCot[lambda2*y]+b3*ArcCot[lambda3*z] ) * w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \left (\text {lambda1}^2 x^2+1\right )^{\frac {\text {b1}}{2 \text {a1} \text {lambda1}}} \left (\text {a1}^2 \left (\text {lambda2}^2 y^2+1\right )\right )^{\frac {\text {b2}}{2 \text {a2} \text {lambda2}}} \left (\text {a1}^2 \left (\text {lambda3}^2 z^2+1\right )\right )^{\frac {\text {b3}}{2 \text {a3} \text {lambda3}}} c_1\left (y-\frac {\text {a2} x}{\text {a1}},z-\frac {\text {a3} x}{\text {a1}}\right ) \exp \left (\frac {-\text {a1} \text {a2} \text {b3} z \arctan (\text {lambda3} z)-\text {a1} \text {a3} \text {b2} y \arctan (\text {lambda2} y)+\text {a2} \text {a3} \text {b2} x \arctan (\text {lambda2} y)+\text {a2} \text {a3} \text {b3} x \arctan (\text {lambda3} z)+\text {a2} \text {a3} \text {b1} x \cot ^{-1}(\text {lambda1} x)+\text {a2} \text {a3} \text {b2} x \cot ^{-1}(\text {lambda2} y)+\text {a2} \text {a3} \text {b3} x \cot ^{-1}(\text {lambda3} z)}{\text {a1} \text {a2} \text {a3}}\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  a__1*diff(w(x,y,z),x)+ a__2*diff(w(x,y,z),y)+ a__3*diff(w(x,y,z),z)= (b__1*arccot(lambda__1*x)+b__2*arccot(lambda__2*y)+b__3*arccot(lambda__3*z))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a_{1} y -a_{2} x}{a_{1}}, \frac {a_{1} z -a_{3} x}{a_{1}}\right ) \left (\lambda _{1}^{2} x^{2}+1\right )^{\frac {b_{1}}{2 a_{1} \lambda _{1}}} \left (\lambda _{2}^{2} y^{2}+1\right )^{\frac {b_{2}}{2 \lambda _{2} a_{2}}} \left (\lambda _{3}^{2} z^{2}+1\right )^{\frac {b_{3}}{2 \lambda _{3} a_{3}}} {\mathrm e}^{\frac {\operatorname {arccot}\left (\lambda _{2} y \right ) a_{1} a_{3} b_{2} y +\operatorname {arccot}\left (\lambda _{3} z \right ) a_{1} a_{2} b_{3} z +b_{1} \operatorname {arccot}\left (\lambda _{1} x \right ) x a_{2} a_{3}}{a_{1} a_{2} a_{3}}}\]

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6.8.22.3 [1888] Problem 3

problem number 1888

Added Nov 30, 2019.

Problem Chapter 8.7.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -b x}{a}, -\int \operatorname {arccot}\left (\lambda x \right )^{n}d x +\frac {a \int \operatorname {arccot}\left (\beta z \right )^{-k}d z}{c}\right ) {\mathrm e}^{\frac {s \,\operatorname {arccot}\left (\gamma x \right ) x}{a}} \left (\gamma ^{2} x^{2}+1\right )^{\frac {s}{2 a \gamma }}\]

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6.8.22.4 [1889] Problem 4

problem number 1889

Added Nov 30, 2019.

Problem Chapter 8.7.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

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

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6.8.22.5 [1890] Problem 5

problem number 1890

Added Nov 30, 2019.

Problem Chapter 8.7.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^z\frac {s \cot ^{-1}(\beta K[3])^{-k} \cot ^{-1}\left (\frac {\gamma \left (c x-a \int _1^z\cot ^{-1}(\beta K[2])^{-k}dK[2]+a \int _1^{K[3]}\cot ^{-1}(\beta K[2])^{-k}dK[2]\right )}{c}\right ){}^m}{c}dK[3]\right ) c_1\left (y-\int _1^x\frac {b \cot ^{-1}(\lambda K[1])^n}{a}dK[1],\int _1^z\cot ^{-1}(\beta K[2])^{-k}dK[2]-\frac {c x}{a}\right )\right \}\right \}\]

Maple ✓

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

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

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