6.4.18 6.4

6.4.18.1 [1129] Problem 1

problem number 1129

Added March 9, 2019.

Problem Chapter 4.6.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {c (\log (\tan (\lambda x+\mu y))+\log (\cos (\lambda x+\mu y)))}{a \lambda +b \mu }\right )\right \}\right \}\]

Maple ✓

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

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

____________________________________________________________________________________

6.4.18.2 [1130] Problem 2

problem number 1130

Added March 9, 2019.

Problem Chapter 4.6.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = (c \cot (\lambda x)+ k \cot (\mu y) ) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \sin ^{\frac {c}{a \lambda }}(\lambda x) c_1\left (y-\frac {b x}{a}\right ) e^{\frac {k (\log (\tan (\mu y))+\log (\cos (\mu y)))}{b \mu }}\right \}\right \}\]

Maple ✓

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

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

____________________________________________________________________________________

6.4.18.3 [1131] Problem 3

problem number 1131

Added March 9, 2019.

Problem Chapter 4.6.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a x \cot (\lambda x+ \mu y) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right ) \exp \left (\frac {a x (\log (\tan (\lambda x+\mu y))+\log (\cos (\lambda x+\mu y)))}{\lambda x+\mu y}\right )\right \}\right \}\]

Maple ✓

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

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

____________________________________________________________________________________

6.4.18.4 [1132] Problem 4

problem number 1132

Added March 9, 2019.

Problem Chapter 4.6.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {b \cot ^{n+1}(\lambda x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(\lambda x)\right )}{a \lambda n+a \lambda }+y\right ) \exp \left (\int _1^x\frac {s \cot ^k\left (\frac {\beta \left (b \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(\lambda x)\right ) \cot ^{n+1}(\lambda x)+a \lambda (n+1) y-b \cot ^{n+1}(\lambda K[1]) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(\lambda K[1])\right )\right )}{a \lambda (n+1)}\right )+c \cot ^m(\mu K[1])}{a}dK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (y -\left (\int \frac {b \left (\cot ^{n}\left (\lambda x \right )\right )}{a}d x \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {c \left (\cot ^{m}\left (\mathit {\_b} \mu \right )\right )+s \left (-\cot \left (\left (-y -\left (\int \frac {b \left (\cot ^{n}\left (\mathit {\_b} \lambda \right )\right )}{a}d \mathit {\_b} \right )+\int \frac {b \left (\cot ^{n}\left (\lambda x \right )\right )}{a}d x \right ) \beta \right )\right )^{k}}{a}d\mathit {\_b}}\]

____________________________________________________________________________________

6.4.18.5 [1133] Problem 5

problem number 1133

Added March 9, 2019.

Problem Chapter 4.6.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\cot ^{1-n}(\lambda y) \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};-\cot ^2(\lambda y)\right )}{\lambda (n-1)}-\frac {b x}{a}\right ) \exp \left (\int _1^y\frac {\left (s \cot ^k(\beta K[1])+c \cot ^m\left (\frac {a \mu \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};-\cot ^2(\lambda y)\right ) \cot ^{1-n}(\lambda y)+b \lambda \mu x-b \lambda \mu n x-a \mu \cot ^{1-n}(\lambda K[1]) \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};-\cot ^2(\lambda K[1])\right )}{b \lambda -b \lambda n}\right )\right ) \cot ^{-n}(\lambda K[1])}{b}dK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {a \left (\int \left (\cot ^{-n}\left (\lambda y \right )\right )d y \right )}{b}+x \right ) {\mathrm e}^{\int _{}^{y}\frac {\left (c \left (-\cot \left (-\mu \left (\int \frac {a \left (\cot ^{-n}\left (\mathit {\_b} \lambda \right )\right )}{b}d \mathit {\_b} \right )-\left (-\frac {a \left (\int \left (\cot ^{-n}\left (\lambda y \right )\right )d y \right )}{b}+x \right ) \mu \right )\right )^{m}+s \left (\cot ^{k}\left (\mathit {\_b} \beta \right )\right )\right ) \left (\cot ^{-n}\left (\mathit {\_b} \lambda \right )\right )}{b}d\mathit {\_b}}\]

____________________________________________________________________________________