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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

\[w \left (x , y\right ) = \frac {\int _{}^{y}\frac {a \left (\frac {\pi }{2}-\arcsin \left (\frac {\mu \left (x b \lambda -\operatorname {Si}\left (\arccos \left (\lambda \textit {\_a} \right )\right ) a +\operatorname {Si}\left (\arccos \left (\lambda y \right )\right ) a \right )}{b \lambda }\right )\right )^{m}+\arccos \left (\beta \textit {\_a} \right )^{k}}{\arccos \left (\lambda \textit {\_a} \right )}d \textit {\_a}}{b}+f_{1} \left (\frac {x b \lambda +\operatorname {Si}\left (\arccos \left (\lambda y \right )\right ) a}{b \lambda }\right )\]

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