6.3.19 6.4

6.3.19.1 [951] Problem 1

problem number 951

Added Feb. 11, 2019.

Problem Chapter 3.6.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.19.2 [952] Problem 2

problem number 952

Added Feb. 11, 2019.

Problem Chapter 3.6.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.19.3 [953] Problem 3

problem number 953

Added Feb. 11, 2019.

Problem Chapter 3.6.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.19.4 [954] Problem 4

problem number 954

Added Feb. 11, 2019.

Problem Chapter 3.6.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.3.19.5 [955] Problem 5

problem number 955

Added Feb. 11, 2019.

Problem Chapter 3.6.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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