6.8.20 7.2

6.8.20.1 [1875] Problem 1

problem number 1875

Added Nov 30, 2019.

Problem Chapter 8.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \arccos (\beta x)^n \left (\arccos (\beta x)^2\right )^{-n} \left ((-i \arccos (\beta x))^n \Gamma (n+1,i \arccos (\beta x))+(i \arccos (\beta x))^n \Gamma (n+1,-i \arccos (\beta x))\right )}{2 \beta }\right ) c_1(y-a x,z-b x)\right \}\right \}\]

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

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

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6.8.20.2 [1876] Problem 2

problem number 1876

Added Nov 30, 2019.

Problem Chapter 8.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*ArcCos[lambda1*x]+b2*ArcCos[lambda2*y]+b3*ArcCos[lambda3*z] ) * w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

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*arccos(lambda__1*x)+b__2*arccos(lambda__2*y)+b__3*arccos(lambda__3*z))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.8.20.3 [1877] Problem 3

problem number 1877

Added Nov 30, 2019.

Problem Chapter 8.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -x b}{a}, \frac {-\sqrt {\arccos \left (\beta z \right )}\, \left (\left (-2-n \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )+\arccos \left (\lambda x \right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right )-\arccos \left (\lambda x \right )^{n +\frac {3}{2}}\right ) \left (-2+k \right ) \beta c \sqrt {-\lambda ^{2} x^{2}+1}+\lambda \left (2+n \right ) \left (\sqrt {\arccos \left (\lambda x \right )}\, \left (\left (2-k \right ) \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )-\operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )+\arccos \left (\beta z \right )^{-k +\frac {3}{2}}\right ) a \sqrt {-\beta ^{2} z^{2}+1}+\beta \left (-2+k \right ) \left (a \sqrt {\arccos \left (\lambda x \right )}\, \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) z \arccos \left (\beta z \right )-\arccos \left (\lambda x \right ) \sqrt {\arccos \left (\beta z \right )}\, c \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right ) x \right )\right )}{\sqrt {\arccos \left (\lambda x \right )}\, \sqrt {\arccos \left (\beta z \right )}\, \lambda \left (2+n \right ) c \beta \left (-2+k \right )}\right ) {\mathrm e}^{\frac {s \left (\gamma x \arccos \left (\gamma x \right )-\sqrt {-\gamma ^{2} x^{2}+1}\right )}{a \gamma }}\]

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6.8.20.4 [1878] Problem 4

problem number 1878

Added Nov 30, 2019.

Problem Chapter 8.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+c*ArcCos[lambda*x]^n*ArcCos[beta*y]^m*ArcCos[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*arccos(lambda*x)^n*arccos(beta*y)^m*arccos(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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -x b}{a}, \frac {-\gamma \sqrt {\arccos \left (\gamma z \right )}\, c \left (-2+k \right ) \int _{}^{x}\arccos \left (\lambda \textit {\_a} \right )^{n} \arccos \left (\frac {\beta \left (a y -b \left (x -\textit {\_a} \right )\right )}{a}\right )^{m}d \textit {\_a} +\left (\left (\left (2-k \right ) \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\gamma z \right )\right )-\arccos \left (\gamma z \right ) \operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\gamma z \right )\right )+\arccos \left (\gamma z \right )^{-k +\frac {3}{2}}\right ) \sqrt {-\gamma ^{2} z^{2}+1}+\gamma \arccos \left (\gamma z \right ) \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\gamma z \right )\right ) z \left (-2+k \right )\right ) a}{\sqrt {\arccos \left (\gamma z \right )}\, c \gamma \left (-2+k \right )}\right ) {\mathrm e}^{\frac {s x}{a}}\]

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6.8.20.5 [1879] Problem 5

problem number 1879

Added Nov 30, 2019.

Problem Chapter 8.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[\text {Expression too large to display}\]

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6.8.20.6 [1880] Problem 6

problem number 1880

Added Nov 30, 2019.

Problem Chapter 8.7.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*D[w[x,y,z],x]+b*ArcCos[lambda*y]^n*D[w[x,y,z],y]+c*ArcCos[beta*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*arccos(lambda*y)^n*diff(w(x,y,z),y)+ c*arccos(beta*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 {\left (\left (2-n \right ) \operatorname {LommelS1}\left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda y \right )\right )-\operatorname {LommelS1}\left (-n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda y \right )\right ) \arccos \left (\lambda y \right )+\arccos \left (\lambda y \right )^{-n +\frac {3}{2}}\right ) a \sqrt {-\lambda ^{2} y^{2}+1}+\lambda \left (-2+n \right ) \left (a \operatorname {LommelS1}\left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda y \right )\right ) y \arccos \left (\lambda y \right )-\sqrt {\arccos \left (\lambda y \right )}\, b x \right )}{\sqrt {\arccos \left (\lambda y \right )}\, b \lambda \left (-2+n \right )}, \frac {\beta \left (-2+k \right ) c \left (\left (-2+n \right ) \operatorname {LommelS1}\left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda y \right )\right )+\operatorname {LommelS1}\left (-n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda y \right )\right ) \arccos \left (\lambda y \right )-\arccos \left (\lambda y \right )^{-n +\frac {3}{2}}\right ) \sqrt {\arccos \left (\beta z \right )}\, \sqrt {-\lambda ^{2} y^{2}+1}+\left (-2+n \right ) \left (\left (\left (2-k \right ) \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )-\operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )+\arccos \left (\beta z \right )^{-k +\frac {3}{2}}\right ) \sqrt {\arccos \left (\lambda y \right )}\, b \sqrt {-\beta ^{2} z^{2}+1}+\beta \left (-2+k \right ) \left (b \sqrt {\arccos \left (\lambda y \right )}\, \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) z \arccos \left (\beta z \right )-\arccos \left (\lambda y \right ) \sqrt {\arccos \left (\beta z \right )}\, c y \operatorname {LommelS1}\left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda y \right )\right )\right )\right ) \lambda }{\sqrt {\arccos \left (\lambda y \right )}\, \sqrt {\arccos \left (\beta z \right )}\, \lambda \left (-2+n \right ) c \beta \left (-2+k \right )}\right ) {\mathrm e}^{\frac {s \int \arccos \left (\lambda y \right )^{-n}d y}{b}}\]

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