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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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