6.3.22 7.2

6.3.22.1 [967] Problem 1

problem number 967

Added Feb. 11, 2019.

Problem Chapter 3.7.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \arccos \frac {x}{\lambda }+ k \arccos \frac {y}{\beta } \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \frac {b c x \arccos \left (\frac {x}{\lambda }\right )+a b \mathit {\_F1} \left (\frac {a y -b x}{a}\right )-\sqrt {-\frac {x^{2}}{\lambda ^{2}}+1}\, b c \lambda -\left (-y \arccos \left (\frac {y}{\beta }\right )+\sqrt {\frac {\beta ^{2}-y^{2}}{\beta ^{2}}}\, \beta \right ) a k}{a b}\]

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6.3.22.2 [968] Problem 2

problem number 968

Added Feb. 11, 2019.

Problem Chapter 3.7.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \frac {\left (\beta y +\lambda x \right ) c \arccos \left (\beta y +\lambda x \right )-\sqrt {-\beta ^{2} y^{2}-2 \beta \lambda x y -\lambda ^{2} x^{2}+1}\, c +\left (a \lambda +b \beta \right ) \mathit {\_F1} \left (\frac {a y -b x}{a}\right )}{a \lambda +b \beta }\]

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6.3.22.3 [969] Problem 3

problem number 969

Added Feb. 11, 2019.

Problem Chapter 3.7.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \frac {-\sqrt {-\left (\frac {\beta y}{x}+\lambda \right )^{2} x^{2}+1}\, a x +\left (\beta y +\lambda x \right ) \left (a x \arccos \left (\beta y +\lambda x \right )+\mathit {\_F1} \left (\frac {y}{x}\right )\right )}{\beta y +\lambda x}\]

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6.3.22.4 [970] Problem 4

problem number 970

Added Feb. 11, 2019.

Problem Chapter 3.7.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \int _{}^{x}\left (\arccos \left (\mathit {\_a} \mu \right )^{m}+\frac {\arccos \left (\frac {\left (-\sqrt {-\mathit {\_a}^{2} \lambda ^{2}+1}\, b +\sqrt {-\lambda ^{2} x^{2}+1}\, b +\left (\mathit {\_a} b \arccos \left (\mathit {\_a} \lambda \right )-b x \arccos \left (\lambda x \right )+a y \right ) \lambda \right ) \beta }{a \lambda }\right )^{k}}{a}\right )d\mathit {\_a} +\mathit {\_F1} \left (\frac {-b \lambda x \arccos \left (\lambda x \right )+a \lambda y +\sqrt {-\lambda ^{2} x^{2}+1}\, b}{a \lambda }\right )\]

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6.3.22.5 [971] Problem 5

problem number 971

Added Feb. 11, 2019.

Problem Chapter 3.7.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \int _{}^{y}\frac {a \arccos \left (\frac {\left (b \lambda x -a \Si \left (\arccos \left (\mathit {\_a} \lambda \right )\right )+a \Si \left (\arccos \left (\lambda y \right )\right )\right ) \mu }{b \lambda }\right )^{m}+\arccos \left (\mathit {\_a} \beta \right )^{k}}{b \arccos \left (\mathit {\_a} \lambda \right )}d\mathit {\_a} +\mathit {\_F1} \left (\frac {b \lambda x +a \Si \left (\arccos \left (\lambda y \right )\right )}{b \lambda }\right )\]

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