6.6.21 7.3

6.6.21.1 [1540] Problem 1

problem number 1540

Added May 31, 2019.

Problem Chapter 6.7.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*arctan(lambda*x)^n*arctan(beta*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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},-\int \! \left ( \arctan \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+\int \!{\frac { \left ( \arctan \left ( \beta \,z \right ) \right ) ^{-k}a}{c}}\,{\rm d}z \right ) \]

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6.6.21.2 [1541] Problem 2

problem number 1541

Added May 31, 2019.

Problem Chapter 6.7.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\int _1^z\tan ^{-1}(\gamma K[1])^{-k}dK[1]-\int _1^x\frac {c \tan ^{-1}(\lambda K[2])^n \left (\left (\frac {a \tan ^{-1}(\lambda K[2])^{-n} \text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\int _1^x\frac {c \tan ^{-1}(\lambda K[2])^n \tan ^{-1}\left (\beta \left (y+\frac {b (K[2]-x)}{a}\right )\right )^m}{a}dK[2],\{K[2],1,x\}\right ]}{c}\right ){}^{\frac {1}{m}}\right ){}^m}{a}dK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*arctan(lambda*x)^n*arctan(beta*y)^m*arctan(gamma1*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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},-\int ^{x}\! \left ( \arctan \left ( {\it \_a}\,\lambda \right ) \right ) ^{n} \left ( \arctan \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{m}{d{\it \_a}}+\int \!{\frac { \left ( \arctan \left ( \gamma 1\,z \right ) \right ) ^{-k}a}{c}}\,{\rm d}z \right ) \]

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6.6.21.3 [1542] Problem 3

problem number 1542

Added May 31, 2019.

Problem Chapter 6.7.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*arctan(lambda*x)^n*diff(w(x,y,z),y)+c*arctan(beta*x)^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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!{\frac {b \left ( \arctan \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x+y,-\int \!{\frac {c \left ( \arctan \left ( \beta \,x \right ) \right ) ^{k}}{a}}\,{\rm d}x+z \right ) \]

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6.6.21.4 [1543] Problem 4

problem number 1543

Added May 31, 2019.

Problem Chapter 6.7.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*arctan(lambda*x)^n*diff(w(x,y,z),y)+c*arctan(beta*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'));
 

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

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6.6.21.5 [1544] Problem 5

problem number 1544

Added May 31, 2019.

Problem Chapter 6.7.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*arctan(lambda*y)^n*diff(w(x,y,z),y)+c*arctan(beta*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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -{\frac {a\int \! \left ( \arctan \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x,-\int \! \left ( \arctan \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y+\int \!{\frac { \left ( \arctan \left ( \beta \,z \right ) \right ) ^{-k}b}{c}}\,{\rm d}z \right ) \]

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