6.3.22 7.2

6.3.22.1 [968] Problem 1

problem number 968

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'));
 

\[w \left ( x,y \right ) ={\frac {1}{ab} \left ( -\sqrt {-{\frac {{x}^{2}}{{\lambda }^{2}}}+1}bc\lambda +{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) ba+\arccos \left ( {\frac {x}{\lambda }} \right ) bcx-k \left ( \sqrt {{\frac {{\beta }^{2}-{y}^{2}}{{\beta }^{2}}}}\beta -y\arccos \left ( {\frac {y}{\beta }} \right ) \right ) a \right ) }\]

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6.3.22.2 [969] Problem 2

problem number 969

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]];
 

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

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'));
 

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

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6.3.22.3 [970] Problem 3

problem number 970

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]];
 

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

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'));
 

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

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6.3.22.4 [971] Problem 4

problem number 971

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]];
 

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

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'));
 

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

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6.3.22.5 [972] Problem 5

problem number 972

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]];
 

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

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'));
 

\[w \left ( x,y \right ) =\int ^{y}\!{\frac {1}{\arccos \left ( {\it \_a}\,\lambda \right ) b} \left ( a \left ( \arccos \left ( {\frac {\mu \, \left ( bx\lambda +\Si \left ( \arccos \left ( y\lambda \right ) \right ) a-a\Si \left ( \arccos \left ( {\it \_a}\,\lambda \right ) \right ) \right ) }{\lambda \,b}} \right ) \right ) ^{m}+ \left ( \arccos \left ( \beta \,{\it \_a} \right ) \right ) ^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {bx\lambda +\Si \left ( \arccos \left ( y\lambda \right ) \right ) a}{\lambda \,b}} \right ) \]

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