6.3.23 7.3

6.3.23.1 [972] Problem 1

problem number 972

Added Feb. 11, 2019.

Problem Chapter 3.7.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = c \arctan \frac {x}{\lambda }+ k \arctan \frac {y}{\beta } \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.23.2 [973] Problem 2

problem number 973

Added Feb. 11, 2019.

Problem Chapter 3.7.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = c \arctan (\lambda x+\beta y) \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.23.3 [974] Problem 3

problem number 974

Added Feb. 11, 2019.

Problem Chapter 3.7.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + y w_y = a x \arctan (\lambda x+\beta y) \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.23.4 [975] Problem 4

problem number 975

Added Feb. 11, 2019.

Problem Chapter 3.7.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.23.5 [976] Problem 5

problem number 976

Added Feb. 11, 2019.

Problem Chapter 3.7.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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