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]];
 

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

Maple ✓

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

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

\[\left \{\left \{w(x,y)\to \int _1^x\frac {s \cot ^k\left (\frac {\beta \left (b \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\lambda x)\right ) \cot ^{n+1}(\lambda x)+a \lambda (n+1) y-b \cot ^{n+1}(\lambda K[1]) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\lambda K[1])\right )\right )}{a \lambda (n+1)}\right )+c \cot ^m(\mu K[1])}{a}dK[1]+c_1\left (\frac {b \cot ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\lambda x)\right )}{a \lambda n+a \lambda }+y\right )\right \}\right \}\]

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'));
 

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

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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]];
 

\[\left \{\left \{w(x,y)\to \int _1^y\frac {\left (s \cot ^k(\beta K[1])+c \cot ^m\left (\frac {a \mu \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},-\cot ^2(\lambda y)\right ) \cot ^{1-n}(\lambda y)+b \lambda \mu x-b \lambda \mu n x-a \mu \cot ^{1-n}(\lambda K[1]) \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},-\cot ^2(\lambda K[1])\right )}{b \lambda -b \lambda n}\right )\right ) \cot ^{-n}(\lambda K[1])}{b}dK[1]+c_1\left (\frac {\cot ^{1-n}(\lambda y) \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},-\cot ^2(\lambda y)\right )}{\lambda (n-1)}-\frac {b x}{a}\right )\right \}\right \}\]

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'));
 

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

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