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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac {b \left ( \cot \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( c \left ( \cot \left ( \mu \,{\it \_b} \right ) \right ) ^{m}+s \left ( -\cot \left ( \beta \, \left ( -\int \!{\frac {b \left ( \cot \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}{\it \_b}+\int \!{\frac {b \left ( \cot \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x-y \right ) \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]

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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'));
 

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

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