6.4.22 7.3

6.4.22.1 [1150] Problem 1

problem number 1150

Added March 9, 2019.

Problem Chapter 4.7.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.22.2 [1151] Problem 2

problem number 1151

Added March 9, 2019.

Problem Chapter 4.7.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\int _1^x\frac {c \arctan \left (\beta y+\lambda K[1]+\frac {b \beta (K[1]-x)}{a}\right )}{a}dK[1]\right )\right \}\right \}\]

Maple ✓

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

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

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6.4.22.3 [1152] Problem 3

problem number 1152

Added March 9, 2019.

Problem Chapter 4.7.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\int _1^x\arctan \left (\beta y+\lambda K[1]+\frac {b \beta (K[1]-x)}{a}\right ) K[1]dK[1]\right )\right \}\right \}\]

Maple ✓

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

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

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6.4.22.4 [1153] Problem 4

problem number 1153

Added March 9, 2019.

Problem Chapter 4.7.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.22.5 [1154] Problem 5

problem number 1154

Added March 9, 2019.

Problem Chapter 4.7.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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