6.6.20 7.2
6.6.20.1 [1535] Problem 1
problem number 1535
Added May 31, 2019.
Problem Chapter 6.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \arccos ^n(\lambda x) \arccos ^k(\beta z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*ArcCos[lambda*x]^n*ArcCos[beta*z]^k*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\int _1^z\arccos (\beta K[1])^{-k}dK[1]-\int _1^x\frac {c \arccos (\lambda K[2])^n}{a}dK[2]\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*arccos(lambda*x)^n*arccos(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'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -b x}{a}, \frac {-\sqrt {\arccos \left (\beta z \right )}\, \beta c \left (\left (-n -2\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )+\arccos \left (\lambda x \right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right )-\arccos \left (\lambda x \right )^{n +\frac {3}{2}}\right ) \left (-2+k \right ) \sqrt {-\lambda ^{2} x^{2}+1}+\lambda \left (\left (\left (2-k \right ) \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )-\operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )+\arccos \left (\beta z \right )^{-k +\frac {3}{2}}\right ) a \sqrt {\arccos \left (\lambda x \right )}\, \sqrt {-\beta ^{2} z^{2}+1}+\beta \left (-2+k \right ) \left (a \sqrt {\arccos \left (\lambda x \right )}\, \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) z \arccos \left (\beta z \right )-\arccos \left (\lambda x \right ) \sqrt {\arccos \left (\beta z \right )}\, c \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right ) x \right )\right ) \left (2+n \right )}{\sqrt {\arccos \left (\lambda x \right )}\, \sqrt {\arccos \left (\beta z \right )}\, \lambda \left (2+n \right ) c \beta \left (-2+k \right )}\right )\]
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6.6.20.2 [1536] Problem 2
problem number 1536
Added May 31, 2019.
Problem Chapter 6.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \arccos ^n(\lambda x) \arccos ^m(\beta y) \arccos ^k(\gamma z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*ArcCos[lambda*x]^n*ArcCos[beta*y]^m*ArcCos[gamma*z]^k*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\int _1^z\arccos (\gamma K[1])^{-k}dK[1]-\int _1^x\frac {c \arccos (\lambda K[2])^n \left (\left (\frac {a \arccos (\lambda K[2])^{-n} \text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\int _1^x\frac {c \arccos (\lambda K[2])^n \arccos \left (\beta \left (y+\frac {b (K[2]-x)}{a}\right )\right )^m}{a}dK[2],\{K[2],1,x\}\right ]}{c}\right ){}^{\frac {1}{m}}\right ){}^m}{a}dK[2]\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*arccos(lambda*x)^n*arccos(beta*y)^m*arccos(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'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -b x}{a}, \frac {-\gamma \operatorname {1} \sqrt {\arccos \left (\gamma \operatorname {1} z \right )}\, c \left (-2+k \right ) \int _{}^{x}\arccos \left (\lambda \textit {\_a} \right )^{n} \arccos \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \beta }{a}\right )^{m}d \textit {\_a} +a \left (\left (\left (2-k \right ) \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\gamma \operatorname {1} z \right )\right )-\arccos \left (\gamma \operatorname {1} z \right ) \operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\gamma \operatorname {1} z \right )\right )+\arccos \left (\gamma \operatorname {1} z \right )^{-k +\frac {3}{2}}\right ) \sqrt {-\gamma \operatorname {1}^{2} z^{2}+1}+\gamma \operatorname {1} \arccos \left (\gamma \operatorname {1} z \right ) \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\gamma \operatorname {1} z \right )\right ) z \left (-2+k \right )\right )}{\sqrt {\arccos \left (\gamma \operatorname {1} z \right )}\, c \gamma \operatorname {1} \left (-2+k \right )}\right )\]
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6.6.20.3 [1537] Problem 3
problem number 1537
Added May 31, 2019.
Problem Chapter 6.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \arccos ^n(\lambda x) w_y + c \arccos ^k(\beta x) w_z = 0 \]
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*x]^k*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\int _1^x\frac {b \arccos (\lambda K[1])^n}{a}dK[1],z-\int _1^x\frac {c \arccos (\beta K[2])^k}{a}dK[2]\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y,z),x)+ b*arccos(lambda*x)^n*diff(w(x,y,z),y)+c*arccos(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'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {\left (\left (2+n \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right ) \arccos \left (\lambda x \right )+\arccos \left (\lambda x \right )^{n +\frac {3}{2}}\right ) b \sqrt {-\lambda ^{2} x^{2}+1}+\lambda \left (2+n \right ) \left (-b x \arccos \left (\lambda x \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )+a \sqrt {\arccos \left (\lambda x \right )}\, y \right )}{\sqrt {\arccos \left (\lambda x \right )}\, \left (2+n \right ) a \lambda }, \frac {\left (\left (2+k \right ) \operatorname {LommelS1}\left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta x \right )\right )-\operatorname {LommelS1}\left (k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\beta x \right )\right ) \arccos \left (\beta x \right )+\arccos \left (\beta x \right )^{k +\frac {3}{2}}\right ) c \sqrt {-\beta ^{2} x^{2}+1}+\beta \left (2+k \right ) \left (-c x \arccos \left (\beta x \right ) \operatorname {LommelS1}\left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta x \right )\right )+a \sqrt {\arccos \left (\beta x \right )}\, z \right )}{\sqrt {\arccos \left (\beta x \right )}\, \left (2+k \right ) a \beta }\right )\]
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6.6.20.4 [1538] Problem 4
problem number 1538
Added May 31, 2019.
Problem Chapter 6.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \arccos ^n(\lambda x) w_y + c \arccos ^k(\beta z) w_z = 0 \]
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]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\int _1^x\frac {b \arccos (\lambda K[1])^n}{a}dK[1],\int _1^z\arccos (\beta K[2])^{-k}dK[2]-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart;
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)= 0;
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 x \right )\right )-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right ) \arccos \left (\lambda x \right )+\arccos \left (\lambda x \right )^{n +\frac {3}{2}}\right ) b \sqrt {-\lambda ^{2} x^{2}+1}+\lambda \left (2+n \right ) \left (-b x \arccos \left (\lambda x \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )+a \sqrt {\arccos \left (\lambda x \right )}\, y \right )}{\sqrt {\arccos \left (\lambda x \right )}\, a \lambda \left (2+n \right )}, \frac {-\beta \sqrt {\arccos \left (\beta z \right )}\, c \left (-2+k \right ) \int _{}^{y}{\arccos \left (\lambda \operatorname {RootOf}\left (2 \arccos \left (\lambda \textit {\_Z} \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda \textit {\_Z} \right )\right ) \textit {\_Z} b \lambda n -2 \lambda \sqrt {\arccos \left (\lambda \textit {\_Z} \right )}\, n b \int \arccos \left (\lambda x \right )^{n}d x -2 \textit {\_b} a \lambda \sqrt {\arccos \left (\lambda \textit {\_Z} \right )}\, n +2 a \lambda \sqrt {\arccos \left (\lambda \textit {\_Z} \right )}\, n y +4 \arccos \left (\lambda \textit {\_Z} \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda \textit {\_Z} \right )\right ) \textit {\_Z} b \lambda -4 \lambda \sqrt {\arccos \left (\lambda \textit {\_Z} \right )}\, b \int \arccos \left (\lambda x \right )^{n}d x -4 \textit {\_b} a \lambda \sqrt {\arccos \left (\lambda \textit {\_Z} \right )}+4 a \lambda \sqrt {\arccos \left (\lambda \textit {\_Z} \right )}\, y -2 \sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda \textit {\_Z} \right )\right ) b n +2 \sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, \arccos \left (\lambda \textit {\_Z} \right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda \textit {\_Z} \right )\right ) b -4 \sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda \textit {\_Z} \right )\right ) b -2 \sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, \arccos \left (\lambda \textit {\_Z} \right )^{n +\frac {3}{2}} b \right )\right )}^{-n}d \textit {\_b} +\left (\left (\left (2-k \right ) \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )-\arccos \left (\beta z \right ) \operatorname {LommelS1}\left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\beta z \right )\right )+\arccos \left (\beta z \right )^{-k +\frac {3}{2}}\right ) \sqrt {-\beta ^{2} z^{2}+1}+\beta \arccos \left (\beta z \right ) \operatorname {LommelS1}\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) z \left (-2+k \right )\right ) b}{\sqrt {\arccos \left (\beta z \right )}\, \beta \left (-2+k \right ) c}\right )\]
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6.6.20.5 [1539] Problem 5
problem number 1539
Added May 31, 2019.
Problem Chapter 6.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \arccos ^n(\lambda y) w_y + c \arccos ^k(\beta z) w_z = 0 \]
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]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\int _1^y\arccos (\lambda K[1])^{-n}dK[1]-\frac {b x}{a},\int _1^z\arccos (\beta K[2])^{-k}dK[2]-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart;
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)= 0;
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 \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 {-\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 {\left (-2+k \right ) c \sqrt {\arccos \left (\beta z \right )}\, \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 ) \beta \sqrt {-\lambda ^{2} y^{2}+1}+\left (b \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 )}\, \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 \left (-2+n \right )}{\sqrt {\arccos \left (\lambda y \right )}\, \sqrt {\arccos \left (\beta z \right )}\, \lambda \left (-2+n \right ) c \beta \left (-2+k \right )}\right )\]
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