6.4.21 7.2

6.4.21.1 [1145] Problem 1

problem number 1145

Added March 9, 2019.

Problem Chapter 4.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \left ( c \arccos (\frac {x}{\lambda } + k \arccos (\frac {y}{\beta } ) \right ) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == (c*ArcCos[x/lambda] + k*ArcCos[y/beta])*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {-\frac {k \left (\sqrt {a^2 \left (\beta ^2-y^2\right )} \arctan \left (\frac {a y}{\sqrt {a^2 \left (\beta ^2-y^2\right )}}\right ) (a y-b x)+a^2 \left (\beta ^2-y^2\right )\right )}{b \beta \sqrt {1-\frac {y^2}{\beta ^2}}}+a k x \arccos \left (\frac {y}{\beta }\right )+a c x \arccos \left (\frac {x}{\lambda }\right )-a c \lambda \sqrt {1-\frac {x^2}{\lambda ^2}}}{a^2}\right )\right \}\right \}\]

Maple ✓

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

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

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6.4.21.2 [1146] Problem 2

problem number 1146

Added March 9, 2019.

Problem Chapter 4.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \arccos (\lambda x+\beta y) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {c \left (x (a \lambda +b \beta ) \arccos (\beta y+\lambda x)+\beta (b x-a y) \arcsin (\beta y+\lambda x)+a \left (-\sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1}\right )\right )}{a (a \lambda +b \beta )}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {y a -b x}{a}\right ) {\mathrm e}^{\frac {c \left (\arccos \left (\beta y +\lambda x \right ) \beta y +\arccos \left (\beta y +\lambda x \right ) \lambda x -\sqrt {-\beta ^{2} y^{2}-2 \beta \lambda x y -x^{2} \lambda ^{2}+1}\right )}{a \lambda +b \beta }}\]

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6.4.21.3 [1147] Problem 3

problem number 1147

Added March 9, 2019.

Problem Chapter 4.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = a x \arccos (\lambda x+\beta y) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == a*x*ArcCos[lambda*x + beta*y]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {1}{4} \left (\frac {\left (a^2 \left (2 \beta ^2 y^2+1\right )-4 a b \beta ^2 x y+2 b^2 \beta ^2 x^2\right ) \arcsin (\beta y+\lambda x)-a \sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1} (-3 a \beta y+a \lambda x+4 b \beta x)}{(a \lambda +b \beta )^2}+2 x^2 \arccos (\beta y+\lambda x)\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = a*x*arccos(lambda*x+beta*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {y a -b x}{a}\right ) {\mathrm e}^{-\frac {\left (2 \arccos \left (\beta y +\lambda x \right ) a \,\beta ^{2} y^{2}-2 \arccos \left (\beta y +\lambda x \right ) a \,\lambda ^{2} x^{2}-4 \arccos \left (\beta y +\lambda x \right ) b \,\beta ^{2} x y -4 \arccos \left (\beta y +\lambda x \right ) b \beta \lambda \,x^{2}-3 \sqrt {-\beta ^{2} y^{2}-2 \beta \lambda x y -x^{2} \lambda ^{2}+1}\, a \beta y +\sqrt {-\beta ^{2} y^{2}-2 \beta \lambda x y -x^{2} \lambda ^{2}+1}\, a \lambda x +4 \sqrt {-\beta ^{2} y^{2}-2 \beta \lambda x y -x^{2} \lambda ^{2}+1}\, b \beta x -\arcsin \left (\beta y +\lambda x \right ) a \right ) a}{4 \left (a \lambda +b \beta \right )^{2}}}\]

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6.4.21.4 [1148] Problem 4

problem number 1148

Added March 9, 2019.

Problem Chapter 4.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arccos ^n(\lambda x)w_y = \left ( c \arccos ^m(\mu x) + s \arccos ^k(\beta y) \right ) w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.21.5 [1149] Problem 5

problem number 1149

Added March 9, 2019.

Problem Chapter 4.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arccos ^n(\lambda y)w_y = \left ( c \arccos ^m(\mu x) + s \arccos ^k(\beta y) \right ) w \]

Mathematica ✓

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

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

Maple ✓

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

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

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