6.7.21 7.2

6.7.21.1 [1707] Problem 1

problem number 1707

Added June 26, 2019.

Problem Chapter 7.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y + b w_z = c \arccos ^k(\lambda x)+s \]

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[lambda*x]^k+s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

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(lambda*x)^k+s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = c \int \arccos \left (\lambda x \right )^{k}d x +s x +f_{1} \left (-a x +y , -b x +z \right )\]

Answer contains unresolved integrals

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6.7.21.2 [1708] Problem 2

problem number 1708

Added June 26, 2019.

Problem Chapter 7.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

Maple ✓

restart; 
local gamma; 
pde :=  a1*diff(w(x,y,z),x)+ a2*diff(w(x,y,z),y)+ a3*diff(w(x,y,z),z)= b1*arccos(lambda1*x)+b2*arccos(lambda2*y)+b3*arccos(lambda3*z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {-\sqrt {-\lambda \operatorname {1}^{2} x^{2}+1}\, \operatorname {a2} \operatorname {a3} \operatorname {b1} \lambda \operatorname {2} \lambda \operatorname {3} +\lambda \operatorname {1} \left (-\lambda \operatorname {3} \operatorname {a1} \operatorname {a3} \operatorname {b2} \sqrt {-\lambda \operatorname {2}^{2} y^{2}+1}+\lambda \operatorname {2} \left (x \lambda \operatorname {3} \arccos \left (\lambda \operatorname {1} x \right ) \operatorname {a2} \operatorname {a3} \operatorname {b1} +\operatorname {a1} \left (\lambda \operatorname {3} \arccos \left (\lambda \operatorname {2} y \right ) y \operatorname {a3} \operatorname {b2} +\operatorname {a2} \left (\lambda \operatorname {3} \arccos \left (\lambda \operatorname {3} z \right ) z \operatorname {b3} +f_{1} \left (\frac {y \operatorname {a1} -x \operatorname {a2}}{\operatorname {a1}}, \frac {z \operatorname {a1} -x \operatorname {a3}}{\operatorname {a1}}\right ) \lambda \operatorname {3} \operatorname {a3} -\sqrt {-\lambda \operatorname {3}^{2} z^{2}+1}\, \operatorname {b3} \right )\right )\right )\right )}{\lambda \operatorname {1} \operatorname {a1} \operatorname {a2} \lambda \operatorname {2} \operatorname {a3} \lambda \operatorname {3} }\]

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6.7.21.3 [1709] Problem 3

problem number 1709

Added June 26, 2019.

Problem Chapter 7.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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6.7.21.4 [1710] Problem 4

problem number 1710

Added June 26, 2019.

Problem Chapter 7.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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; 
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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6.7.21.5 [1711] Problem 5

problem number 1711

Added June 26, 2019.

Problem Chapter 7.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

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

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