6.4.23 7.4

6.4.23.1 [1155] Problem 1

problem number 1155

Added March 9, 2019.

Problem Chapter 4.7.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.4.23.2 [1156] Problem 2

problem number 1156

Added March 9, 2019.

Problem Chapter 4.7.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.4.23.3 [1157] Problem 3

problem number 1157

Added March 9, 2019.

Problem Chapter 4.7.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.4.23.4 [1158] Problem 4

problem number 1158

Added March 9, 2019.

Problem Chapter 4.7.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.4.23.5 [1159] Problem 5

problem number 1159

Added March 9, 2019.

Problem Chapter 4.7.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (-\frac {a \int \operatorname {arccot}\left (\lambda y \right )^{-n}d y}{b}+x \right ) {\mathrm e}^{\frac {\int _{}^{y}\left (c {\left (\frac {\pi }{2}+\arctan \left (\frac {\mu \left (a \int \operatorname {arccot}\left (\lambda y \right )^{-n}d y -a \int \operatorname {arccot}\left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b} -x b \right )}{b}\right )\right )}^{m}+s \operatorname {arccot}\left (\beta \textit {\_b} \right )^{k}\right ) \operatorname {arccot}\left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b}}{b}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________