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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-bx+z \right ) {{\rm e}^{\int \!c \left ( {\frac {\pi }{2}}-\arctan \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x}}\]

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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]];
 

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

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya_{1}-xa_{2}}{a_{1}}},{\frac {za_{1}-xa_{3}}{a_{1}}} \right ) \left ( {\lambda _{1}}^{2}{x}^{2}+1 \right ) ^{{\frac {b_{1}}{2\,a_{1}\,\lambda _{1}}}} \left ( {y}^{2}{\lambda _{2}}^{2}+1 \right ) ^{{\frac {b_{2}}{2\,\lambda _{2}\,a_{2}}}} \left ( {z}^{2}{\lambda _{3}}^{2}+1 \right ) ^{{\frac {b_{3}}{2\,\lambda _{3}\,a_{3}}}}{{\rm e}^{{\frac {-2\,\arctan \left ( \lambda _{2}\,y \right ) ya_{1}\,a_{3}\,b_{2}+ \left ( -2\,b_{3}\,za_{1}\,\arctan \left ( \lambda _{3}\,z \right ) +x \left ( -2\,b_{1}\,\arctan \left ( \lambda _{1}\,x \right ) +\pi \, \left ( b_{3}+b_{1}+b_{2} \right ) \right ) a_{3} \right ) a_{2}}{2\,a_{1}\,a_{2}\,a_{3}}}}}\]

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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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},-\int \! \left ( {\frac {\pi }{2}}-\arctan \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+\int \!{\frac {a}{c} \left ( {\frac {\pi }{2}}-\arctan \left ( \beta \,z \right ) \right ) ^{-k}}\,{\rm d}z \right ) \left ( {\gamma }^{2}{x}^{2}+1 \right ) ^{{\frac {s}{2\,a\gamma }}}{{\rm e}^{{\frac {sx \left ( \pi -2\,\arctan \left ( \gamma \,x \right ) \right ) }{2\,a}}}}\]

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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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},-\int ^{x}\! \left ( {\frac {\pi }{2}}-\arctan \left ( \lambda \,{\it \_a} \right ) \right ) ^{n} \left ( {\frac {\pi }{2}}-\arctan \left ( {\frac {\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{m}{d{\it \_a}}+\int \!{\frac {a}{c} \left ( {\frac {\pi }{2}}-\arctan \left ( \gamma \,z \right ) \right ) ^{-k}}\,{\rm d}z \right ) {{\rm e}^{{\frac {sx}{a}}}}\]

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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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -y+\int \!{\frac {b}{a} \left ( {\frac {\pi }{2}}-\arctan \left ( \lambda \,x \right ) \right ) ^{n}}\,{\rm d}x,-\int ^{y}\! \left ( {\frac {\pi }{2}}-\arctan \left ( \lambda \,\RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!{\frac {b}{a} \left ( {\frac {\pi }{2}}-\arctan \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}}{d{\it \_b}}-y+\int \!{\frac {b}{a} \left ( {\frac {\pi }{2}}-\arctan \left ( \lambda \,x \right ) \right ) ^{n}}\,{\rm d}x \right ) \right ) \right ) ^{-n}{d{\it \_b}}+\int \!{\frac {b}{c} \left ( {\frac {\pi }{2}}-\arctan \left ( \beta \,z \right ) \right ) ^{-k}}\,{\rm d}z \right ) {{\rm e}^{\int ^{y}\!{\frac {s}{b} \left ( {\frac {\pi }{2}}-\arctan \left ( \gamma \,\RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!{\frac {b}{a} \left ( {\frac {\pi }{2}}-\arctan \left ( \lambda \,{\it \_a} \right ) \right ) ^{n}}{d{\it \_a}}-y+\int \!{\frac {b}{a} \left ( {\frac {\pi }{2}}-\arctan \left ( \lambda \,x \right ) \right ) ^{n}}\,{\rm d}x \right ) \right ) \right ) ^{m} \left ( {\rm arccot} \left (\lambda \,\RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!{\frac {b}{a} \left ( {\frac {\pi }{2}}-\arctan \left ( \lambda \,{\it \_a} \right ) \right ) ^{n}}{d{\it \_a}}-y+\int \!{\frac {b}{a} \left ( {\frac {\pi }{2}}-\arctan \left ( \lambda \,x \right ) \right ) ^{n}}\,{\rm d}x \right ) \right ) \right ) ^{-n}}{d{\it \_b}}}}\]

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